S Mantion

A Torres Dias

M Madami

S Tacchi

Nicolas Biziere
email: nicolas.biziere@cemes.fr

Reconfigurable spin wave modes in a Heusler magnonic crystal

We report on the field evolution of the microwave spin wave modes in magnonic crystals made of square antidot lattices etched in a thin film of Co2MnSi Heusler alloy, and having a lateral size s of 200 nm and an edge-to-edge separation d of 600 or 800 nm. The spin wave modes are investigated combining ferromagnetic resonance, Brillouin light scattering experiments and micromagnetic simulations of the static and dynamic magnetic states as a function of the applied field. We show that for applied fields ≥ 10 mT, when the magnetization is quasi-saturated across the crystals, the two samples show similar behavior and the two most intense modes in the spin waves spectra correspond to a mode extending in the channels comprised between the antidot and a mode localized in the region between the antidots. For smaller fields we observe a difference of the measured spectra as a function of d. This is attributed to the fact that for d = 800 nm the magnetization remains partially uniform in the channel between the antidots due to the Heusler cubic crystal anisotropy, while for d = 600 nm the magnetic edge domains around the antidots spread into the channels. Moreover, this anisotropy also allows different remanent states when initializing the crystals with a small magnetic field along two perpendicular directions. This leads to an efficient excitation or extinction of the spin wave modes, making possible to achieve reconfigurable microwave devices exploiting an antidot geometry in a metallic ferromagnetic with low damping coefficient.

I. Introduction.

Magnonics is the field consisting in exciting, manipulating and detecting spin waves (SWs) in a magnetic medium. Potential applications are extremely broad, among which information and beyond CMOS technologies, neuromorphic computing, logic devices, passive components, magnetic sensors [1][START_REF] Chumak | Magnon spintronics[END_REF][START_REF] Mahmoud | Introduction to spin wave computing[END_REF][START_REF] Lenk | The building blocks of magnonics[END_REF][START_REF] Chumak | Magnon transistor for all-magnon data processing[END_REF][START_REF] Schneider | Realization of spin-wave logic gates[END_REF][START_REF] Wang | Realization of a nanoscale magnonic directional coupler for all-magnon circuits[END_REF][START_REF] Wang | A magnonic directional coupler for integrated magnonic half-adders[END_REF][START_REF] Fischer | Experimental prototype of a spin-wave majority gate[END_REF][START_REF] Arai | Neural-Network Computation Using Spin-Wave-Coupled Spin-Torque Oscillators[END_REF] . In particular, SWs in ferrimagnets, ferromagnets and antiferromagnets offer many advantages. For example, their frequencies can range from few hundreds of MHz to THz for wave vectors ranging from millimeters to nanometers [START_REF] Kampfrath | Coherent terahertz control of antiferromagnetic spin waves[END_REF][START_REF] Zhang | Gate-tunable spin waves in antiferromagnetic atomic bilayers[END_REF] . This offers good perspectives to develop low energy consumption devices for on-chip integration. Moreover, the magnetic nature of the medium allows for long life time applications, frequency or wavevector tuning with an external magnetic field or reconfigurable operations [START_REF] Sarker | Reconfigurable magnon interference by on-chip dynamic wavelength conversion[END_REF] among other advantages.

Magnonic crystals (MCs) are a class of magnonic devices consisting of magnetic materials with a periodic modulation of one or several magnetic properties such as magnetization, internal field or anisotropy. This can be achieved by a modulation of the geometrical parameters (dots and antidots 14- 17 , thickness or width modulation in thin films [START_REF] Chumak | Scattering of backward spin waves in a one-dimensional magnonic crystal[END_REF][START_REF] Chumak | Spin-wave propagation in a microstructured magnonic crystal[END_REF] ), the magnetic parameters (bicomponent [START_REF]Active Control of Dipole-Exchange Coupled Magnon Modes in Nanoscale Bicomponent Magnonic Crystals[END_REF][START_REF] Adhikari | Reconfigurable High-Speed Magnonics in Nanoscale Bicomponent Magnonic Crystals with Interface Exchange Interaction[END_REF][START_REF] Tacchi | Forbidden Band Gaps in the Spin-Wave Spectrum of a Two-Dimensional Bicomponent Magnonic Crystal[END_REF][START_REF] Szulc | Reconfigurable Magnonic Crystals Based on Imprinted Magnetization Textures in Hard and Soft Dipolar-Coupled Bilayers[END_REF] or ion implanted structures [START_REF] Obry | A micro-structured ionimplanted magnonic crystal[END_REF] ), the applied magnetic field [START_REF] Chumak | A currentcontrolled, dynamic magnonic crystal[END_REF] , the local strain [START_REF] Grachev | Strainmediated tunability of spin-wave spectra in the adjacent magnonic crystal stripes with piezoelectric layer[END_REF] , the magnetic texture [START_REF] Kaffash | Nanomagnonics with artificial spin ice[END_REF][START_REF] Zakeri | Magnonic crystals: towards terahertz frequencies[END_REF] . Then, MCs offer a great playground for fundamental studies of SWs in confined structures and particularly interesting phenomena such as SWs stopbands, also called frequency band-gaps [START_REF] Gubbiotti | Brillouin light scattering studies of planar metallic magnonic crystals[END_REF][START_REF] Gubbiotti | Brillouin light scattering studies of planar metallic magnonic crystals[END_REF][START_REF] Polushkin | Current-induced distortion of the band structure and formation of pseudogaps in magnonic crystals[END_REF][START_REF] Wang | Design of nanostrip magnonic crystal waveguides with a single magnonic band gap[END_REF] .

MCs is now an intense field of research and already some devices have been demonstrated for microwave filtering, delay lines or data processing [START_REF] Chumak | Magnonic crystals for data processing[END_REF][START_REF] Metaxas | Sensing magnetic nanoparticles using nano-confined ferromagnetic resonances in a magnonic crystal[END_REF][START_REF] Bankowski | Magnonic crystal as a delay line for low-noise auto-oscillators[END_REF] . Also, they are very relevant to make passive microwave devices because their band-gaps and operating frequencies can be easily tuned by the geometrical parameters of the lattice. For components such as microwave filters, the properties sought for the materials are i) low damping coefficient to get long SWs propagation distance and ii) high magnetization values to operate at frequencies up to few GHz (notably for telecommunications technologies) and reach large frequency bandgaps [START_REF] Zivieri | Bragg diffraction of spin waves from a two-dimensional antidot lattice[END_REF][START_REF] Krawczyk | Planewave theory of three-dimensional magnonic crystals[END_REF] . For nanoscale integration, the device is expected as well to work at remanence, which means without the need of permanent magnets, and to be compatible with nano fabrication processes. Finally, the search for reconfigurable operations at remanence, i.e. the possibility to obtain on demand different microwave responses, has become highly desirable to improve the functionalities of the devices [START_REF] Haldar | Functional nanostructures for bias-magnet-free and reconfigurable microwave magnetic devices[END_REF][START_REF] Haldar | Functional magnetic waveguides for magnonics[END_REF][START_REF] Sadovnikov | Reconfigurable 3D magnonic crystal: Tunable and localized spin-wave excitations in CoFeB meandershaped film[END_REF][START_REF]Making a Reconfigurable Artificial Crystal by Ordering Bistable Magnetic Nanowires[END_REF][START_REF]Cubic Anisotropy for a Reconfigurable Magnonic Crystal Based on ${\mathrm{Co}}_{2}\mathrm{Mn}\mathrm{Si} $ Heusler Alloy[END_REF][START_REF] Haldar | Reconfigurable and self-biased magnonic metamaterials[END_REF][START_REF] Grundler | Reconfigurable magnonics heats up[END_REF] .

The Co2MnSi (CMS) full Heusler alloy is very attractive to fulfill all these requirements. In particular, it is a half-metal ferromagnet leading to a small Gilbert-like damping coefficient α below 10 -3 [START_REF] Guillemard | Ultra-low magnetic damping in Co2Mnbased Heusler compounds: promising materials for spintronic[END_REF][START_REF] Singh | Co2MnSi Heusler alloy thin films[END_REF][START_REF] Guillemard | Engineering Co2MnAlxSi1-x Heusler Compounds as a Model System to Correlate Spin Polarization, Intrinsic Gilbert Damping, and Ultrafast Demagnetization[END_REF][START_REF] De Melo | Unveiling transport properties of Co2MnSi Heusler epitaxial thin films with ultra-low magnetic damping[END_REF] . Even if this is still one or two orders of magnitude higher than YIG ferrites, this material is also interesting for its high magnetization value (≈ 5 µB/f.u, µ0Ms ≈ 1.3 T) [START_REF] Abdallah | Evolution of magnetic properties and damping coefficient of Co 2 MnSi Heusler alloy with Mn/Si and Co/Mn atomic disorder[END_REF][START_REF] Neggache Amina | Propriétés électroniques des alliages d'Heusler Co1.5Fe1.5Ge et Co2MnSi[END_REF] and spin polarization, above 90% experimentally [START_REF] Guillemard | Engineering Co2MnAlxSi1-x Heusler Compounds as a Model System to Correlate Spin Polarization, Intrinsic Gilbert Damping, and Ultrafast Demagnetization[END_REF][START_REF] Andrieu | Direct evidence for minority spin gap in the Co2MnSi Heusler compound[END_REF][START_REF] Guillemard | Polycrystalline Co 2 Mn-based Heusler thin films with high spin polarization and low magnetic damping[END_REF] allowing to combine magnonic and spintronic functionalities [START_REF] Chumak | Magnon spintronics[END_REF][START_REF] Demidov | Spin-orbit-torque magnonics[END_REF] . In addition, CMS shows a high Curie temperature (around 985 K) [START_REF]Exchange energies, Curie temperatures and magnons in Heusler compounds -IOPscience[END_REF][START_REF] Kübler | Ab initio estimates of the Curie temperature for magnetic compounds[END_REF] and a relatively high cubic crystal anisotropy (µ0Hk ≈ 20-30 mT) which was predicted to allow reconfigurable SW modes at remanence in antidot lattices [START_REF] Abdallah | Evolution of magnetic properties and damping coefficient of Co 2 MnSi Heusler alloy with Mn/Si and Co/Mn atomic disorder[END_REF][START_REF] Ortiz | Growth, structural, and magnetic characterization of epitaxial Co2MnSi films deposited on MgO and Cr seed layers[END_REF] . However, to our knowledge, only few magnonics studies have been performed up to now with such materials and few Co-based Heusler alloys magnonic crystals exist in the literature [START_REF] Sebastian | Lowdamping spin-wave propagation in a microstructured Co 2 Mn 0[END_REF][START_REF] Stückler | Ultrabroadband spin-wave propagation in Co 2 ( Mn 0.6 Fe 0.4 ) Si thin films[END_REF][START_REF] Wang | Linear and nonlinear spin-wave dynamics in ultralowdamping microstructured Co2FeAl Heusler waveguide[END_REF][START_REF] Sebastian | Nonlinear Emission of Spin-Wave Caustics from an Edge Mode of a Microstructured Co 2 Mn 0[END_REF][START_REF] Pirro | Non-Gilbert-damping Mechanism in a Ferromagnetic Heusler Compound Probed by Nonlinear Spin Dynamics[END_REF][START_REF] Mallick | Tunability of Domain Structure and Magnonic Spectra in Antidot Arrays of Heusler Alloy[END_REF][START_REF] Langer | Parameter-free determination of the exchange constant in thin films using magnonic patterning[END_REF] .

In this work, we report a systematic investigation of the SW modes in two magnonic crystals made of squared antidot lattices, etched from a thin CMS film, and having a lateral size s = 200 nm and an edge-to-edge separation d = 600 or 800 nm. The evolution of the SW modes as a function of the intensity of the external magnetic field applied along a side of the antidot arrays, was studied combining broadband ferromagnetic resonance and micro-focused Brillouin light scattering (micro-BLS) measurements. Micromagnetic simulations of the SWs frequencies at different applied field values are found to be in very good agreement with the measurements of SWs spectra for both crystals, even at low fields for which the magnetization shows nonuniformities. In addition, micro-BLS mapping of the main SWs modes confirms the spatial profiles obtained in the simulations for the modes with the largest intensity. Finally, for d = 800 nm, we show that the cubic anisotropy allows to stabilize quasi-uniform remanent states when the field is applied in two directions perpendicular to each other, leading to a very pronounced extinction of the SW modes for reconfigurable operations.

II. Methods.

A thin film of CMS with a thickness of 45 +/-2 nm has been deposited on a (100) oriented MgO crystal substrate by magnetron sputtering with a base pressure of 2 x 10 -8 mbar. The layer is deposited at 500°C with a speed of about 1 nm/min and annealed at 700° for two hours. This procedure favors the L21 order of the CMS [START_REF] Ortiz | Growth, structural, and magnetic characterization of epitaxial Co2MnSi films deposited on MgO and Cr seed layers[END_REF][START_REF] Ortiz | Broadband Ferromagnetic Resonance Study of Co2MnSi Thin Films: Effect of the Film Thickness[END_REF][START_REF] Abdallah | Evolution of magnetic properties and damping coefficient of Co 2 MnSi Heusler alloy with Mn/Si and Co/Mn atomic disorder[END_REF] . This reference sample is then cut into several pieces. The antidot lattices are then realized in two steps. First the lattice of squares is patterned in a mr-POS EBR 0.3 resist by electron beam lithography. The dimension of the lattice is 70 x 70 µm². Then, the pattern is transferred in the CMS film by Ar + ion beam etching in a dedicated ion beam etching chamber. An example of a square antidot pattern is shown in Fig. 1(a). The nominal lateral size s is 200 +/-10 nm and the separation d is 600 +/-10 nm and 800 +/-10 nm for the first and second sample, respectively. Let's note that the antidots have more rounded edges for the sample with d = 800 nm (images not shown here). Once the antidot lattice is made, it is separated from the rest of the CMS layer by Ar + ion etching all around the MC on an area of about 800 x 800 µm². Finally, a micro-antenna is deposited on top of the MC. Two kinds of antenna's, one for ferromagnetic resonance (FMR) and one for micro-BLS experiments, were realized by LASER UV-lithography followed by a lift-off of Ti10nm/Cu200nm/Au30nm.

Broadband FMR experiments were performed to measure the evolution of the resonance frequencies as a function of the external magnetic field. They are performed using the micro antenna shown in Fig. 1(c) corresponding to a portion of a coplanar waveguide shorted at one end. The micro antenna covers the surface of the MC and is insulated from the MC by a ≈ 200 nm thick Si3N4 passivation layer deposited by magnetron sputtering. The width of the central and outer conductors is win = 30 and wout =15 µm respectively and the gap is g = 5 µm.

The large central conductor allows for a microwave pumping field hRF with a very small k vector such as kmax ≈ 2π/win = 0.209 rad.μm -1 . In order to evaluate the frequency shift induced by the non-zero wavevector with respect to the FMR mode (i.e k = 0 rad.μm -1 ), we calculated the f(k) dispersion curves for stripes of width 600 and 800 nm according to the Slavin-Kalinikos theory [START_REF] Mahmoud | Introduction to spin wave computing[END_REF][START_REF] Haldar | Functional magnetic waveguides for magnonics[END_REF][START_REF] Kalinikos | The dipole-exchange spin wave spectrum for anisotropic ferromagnetic films with mixed exchange boundary conditions[END_REF] . We obtained a maximum shift of about 30 MHz for the mode n = 1 and even less for the modes with a higher quantization number n. Therefore, we considered in the following that we are close to the FMR configuration which normally implies a uniform hRF over the sample.

Two different configurations were used in FMR experiments which are referred as fieldsweep or frequency-sweep FMR measurement in the following. In field-sweep FMR experiments, the external magnetic field H0 is swept from 300 mT to 0 mT while keeping the frequency of hRF constant. The antenna is connected with a non-magnetic microwave picoprobe (Southwest Factory) to an Agilent microwave source (0.1-30 GHz). The microwave reflected signal from the antenna passes through a circulator and a microwave diode connected to a lock-in amplifier. The excited SWs modes are then detected by synchronous detection as the variation of microwave power absorbed by the MC at the resonance field. For this, the applied field H0 is modulated with an amplitude of 0.2 mT at a frequency of 113 Hz.

For frequency-sweep FMR measurements, the microwave frequency ωRF/2π of the pumping field is swept while keeping a constant H0 value. Here, the antenna is connected to a 0.1-20 GHz Anritsu Vectorial Network Analyzer (VNA). The spin wave spectra is measured as the difference ∆S11 = S11(H0) -S11(ref), where S11(H0) is the reflected signal from the antenna at a particular value of H0 and S11(ref) is the reflected signal measured at µ0H0 = 350 mT for which the spin wave spectrum is above 20 GHz. Frequency and fieldsweep measurements are performed with a microwave power set at -20 dBm on the generator output and H0 is applied either in the x or y directions.

Micro-BLS was exploited to image the intensity of the SW modes. Micro-BLS measurements are performed by focusing a single-mode solid-state laser (with a wavelength of 532 nm) at normal incidence onto the sample surface using an objective with numerical aperture of 0.75, giving a spatial resolution of about 250 nm. A (3+3)-pass tandem Fabry-Perot interferometer is used to analyze the inelastically scattered light. A nanopositioning stage allows us to position the sample with a precision down to 10 nm on all three axes. The stripline antenna shown in the Fig. 1(d) and having a width of 2.5 µm, is fed with an electrical probe station ranging from dc up to 20 GHz to excite SWs. The antenna is isolated from the MC by a ≈ 150 nm thick Si3N4 passivation layer (light blue in Fig. 1d), while the rest of the MC is passivated with an only 50 nm thick layer. The microwave power is set at +16 dBm on the microwave generator output. Micro-BLS measurements were carried out in Damon-Eshbach (DE) geometry applying a magnetic field along the y direction, i.e. parallel to the antenna.

In order to explain the evolution of the SWs spectra as a function of H0, we used Mumax3 [START_REF] Vansteenkiste | The design and verification of MuMax3[END_REF] to perform micromagnetic simulations of the equilibrium magnetic states and dynamic SWs modes of the squared antidot lattices with the geometry shown in Fig. 1(b). The unit cell is divided in cubic cells with dimensions 5 x 5 x 5 nm 3 , below the exchange length of the CMS 𝑙 𝑒𝑥 = √2𝐴/µ 0 𝑀 𝑠 2 ≈ 5.4 𝑛𝑚 as defined in the REF [START_REF] Vansteenkiste | The design and verification of MuMax3[END_REF] . To set the magnetic parameters in the simulations, we used the one obtained from FMR measurements (not shown here) performed on a piece of the CMS reference thin film. We obtained the saturation magnetization µ0Ms = 1.28 T, the exchange constant A = 21 x 10 -12 J/m, the cubic crystal anisotropy constant K = 10.1 x 10 3 J/m 3 (µ0Hk ≈ 21 mT), the gyromagnetic ratio γ = 28.8 GHz/T and the Gilbert damping coefficient α = 2 x 10 -3 . Let's note that we measured an extrinsic contribution to the linewidth ∆H0 = 2 mT which is not considered in the simulations. The film thickness is 45 nm. We used 2D periodic boundary conditions (PBC) in order to simulate an infinite MC.

The equilibrium state is calculated at particular values of H0. We apply the field with a 1° misalignment with respect to the symmetry axes (x or y) of the MC to reproduce the uncertainty of alignment for H0 in the experiment. Once the equilibrium state is reached, the system is left to relax during 10 ns defining the initial remanent state used as the starting point for dynamic simulations.

To calculate the frequencies of the SWs modes, we applied a spatially uniform magnetic pulse along the x axis with a temporal Sinc form. The cutting frequency of the pulse is 30 GHz. The amplitude of hRF is 1 mT ensuring to be in the linear regime of SWs excitation. We then recorded 2048 magnetization files with a time step of 10 ps. This leads to a Nyquist frequency of 50 GHz. To obtain the SWs spectra, a Fast Fourier Transform (FFT) is performed on the time varying z component of the magnetization Mz(𝑡) averaged over all cells.

The SW profiles have also been calculated in the plane z = 45 nm (top surface of the MC). To do that, the initial equilibrium remanent state is first subtracted to each temporal magnetic file. Then, FFT and inverse FFT are performed for each cubic cell to respectively filter a particular frequency and recover the corresponding time evolution of the dynamic magnetization component in the direction of hRF, i.e. the mx component. The spin wave profiles shown in Fig. 3, 4 and 8 correspond to the snapshots at the time 𝜏 for which the amplitude of the mx(𝜏) component averaged over all cells is maximum.

III. Results

A. Spin wave modes in

quasi-saturated states.

Fig. 2(a) and (b) present field-sweep FMR measurements for the sample with d = 800 and 600 nm, respectively. H0 is applied along the y direction, parallel to the length of the micro antenna (see Fig. 1c). The intensity of the microwave absorption signal as a function of ωRF/2π and H0 is color coded in grey scale, while top panels show examples of line-cuts of the signals recorded at 15 GHz as a function of H0.

Many absorption peaks, corresponding to the different SWs modes excited by the micro antenna, are observed for both samples. Moreover, the two samples exhibit a similar behavior when µ0H0 ≥ 10 mT (the case 0 ≤ µ0H0 < 10 mT will be discussed in the section III.C). In particular, two main signals, having large amplitude and referred as the mode A and B for the highest and lowest frequency peak respectively, are found in the whole investigated field range. We also observe increasing number of low frequency modes as H0 increases.

These findings are very well reproduced by the micromagnetic simulations, as it can be seen in the Fig. 2 As it can be seen, the calculated spatial profiles of the different SWs modes show a marked evolution as a function of the field amplitude, due to the changes of the equilibrium magnetic state. In particular, two equilibrium states, i.e. the quasi-saturated state for µ0H0 ≥ 50 mT and the intermediate states for 10 mT ≤ µ0H0 < 50 mT can be defined. In the quasi-saturated state, the edge domains around the antidots are very small and show a butterfly shape. Also, the magnetization in the horizontal channels is almost fully saturated in the direction of H0. Instead, in the intermediate state, the size of the edge domains increases spreading into the horizontal channels. In addition, their shape evolves with a rotation of the magnetization along the x direction due to the local shape anisotropy induced by the antidot.

From the comparison between experiments and simulations, we deduce that the mode A observed in the field-sweep FMR measurements corresponds to localized modes (turquoise circle and triangle), having maximum precession amplitude between the antidots. In the quasi-saturated state this mode is characterized by a periodic pattern in the horizontal channels comprised between antidots, having a periodicity of approximately d along the x and y directions. In the intermediate state, this mode exhibits a more complex spatial profile with several oscillation in the horizontal channels. The mode B, instead, corresponds to extended modes (green and yellow circles, yellow triangle) characterized by a quasi-uniform magnetization precession amplitude within the horizontal channels comprised between rows of antidots. The spatial distribution of the magnetization precession for the modes A and B explains their strong FMR signal, because most of the magnetic volume oscillates in phase leading to a large coupling with hRF. Similar extended and localized modes have been reported in previous experiments performed in antidot lattices made of Permalloy for example [START_REF] Neusser | Spinwave localization between nearest and nextnearest neighboring holes in an antidot lattice[END_REF][START_REF] Neusser | Localization, confinement, and fieldcontrolled propagation of spin waves in Ni 80 Fe 20 antidot lattices[END_REF][START_REF] Yu | Dipolar induced, spatially localized resonance in magnetic antidot arrays[END_REF][START_REF] Sklenar | Strongly localized magnetization modes in permalloy antidot lattices[END_REF][START_REF] Pechan | Direct measurement of spatially localized ferromagnetic-resonance modes in an antidot lattice (invited)[END_REF][START_REF] Tacchi | Magnetic Normal Modes in Squared Antidot Array With Circular Holes: A Combined Brillouin Light Scattering and Broadband Ferromagnetic Resonance Study[END_REF][START_REF] Tacchi | Brillouin light scattering studies of 2D magnonic crystals[END_REF] .

It is interesting to notice that the spatial profile of mode B has a more pronounced field dependance as compared to the mode A. Indeed, at high field the mode B shows a nonzero oscillation amplitude in between the antidots, with a π dephasing with respect to the magnetization in the horizontal channel (see the green circle mode for example). When H0 decreases, the oscillation amplitude between the antidots is reduced and the magnetization precession remains in the horizontal channel (yellow circle). Interestingly, such transformation from the green to yellow circle mode occurs at 100 mT and 150 mT for d = 800 and 600 nm, respectively. This can be explained observing that when d decreases, it is more difficult to saturate the magnetization along the y direction between the antidots due to a more important contribution of the local demagnetizing field. Finally, in the intermediate state, one can observe for both samples that the region of quasi-uniform magnetization precession becomes confined in a narrower region comprised between edge domains where the precession amplitude is large and out of phase by π (yellow triangle).

Simulations indicate that for both d values the mode observed at the highest frequency (represented by blue circle and blue triangle modes) in the quasi-saturated state, corresponds to localized SW modes showing a periodic pattern along the x and y directions with a periodicity of approximately d in both directions. In the intermediate state these modes are observed to disappear due to the modification of the static magnetic configuration.

Moreover, simulations reproduce the presence for high field values of additional modes at frequencies below the main peak B. They correspond to the red and violet circles, red square and red star modes, which can be considered as extended modes along the x direction.

The red circle and red star modes can be compared to quantized modes in stripes geometrically defined by the uniformly magnetized region confined between edge domains, showing 3 and 5 antinodes along y, respectively. However, the analogy with stripes is not straightforward since a strong precession is also observed inside (red circle at µ0H0 = 50 mT, red star) or in between (red circle at µ0H0 ≥ 100 mT) the edge domains. Such complex spatial structure leads to a small signal amplitude both experimentally and in the simulations since the spatially averaged dynamic components tend to cancel. For d = 800 nm, when increasing the amplitude of H0 the red circle turns into the red squares mode which shows an even more complex structure along x.

For the mode described by the violet circle, the magnetization precession is strong in a narrow channel localized very closed to the antidots and it becomes more and more confined and localized around the antidots when increasing H0. Such feature explains why its experimental signature becomes less and less visible at high field due to the weak coupling with the antenna. Also, its peculiar spatial profile make this mode very sensitive to geometrical imperfections of the antidots where small variations of shapes and sizes induce large resonance frequencies shifts and broadening of the peaks [START_REF] Mantion | Influence of Ga + milling on the spin waves modes in a Co 2 MnSi Heusler magnonic crystal[END_REF] . Similar arguments can explain the vanishing at high fields of the edge modes localized in the edge domains and represented by the dark blue circle mode in the simulations. In opposite, one can notice that the experimental signal of this mode becomes stronger when decreasing H0. This is because there is a larger magnetization volume that oscillates, not only inside the larger edge domains but also between antidots as shown by the modes described by the dark blue triangle.

Note that the only noticeable difference between d = 800 and 600 nm for the calculated SW spectra in the quasi-saturated state is the presence of an additional mode for d = 800 nm lying between the modes A and B. This mode is described as an orange circle in the Fig. 2(a) and Fig. 3. Simulations reproduce the evolution of its frequency position which tends to be closer from the main peak A (B) at low (high) field. Because of its complex spatial structure, it is not obvious to define whether it is more of an extended or localized mode. Nevertheless, it is also periodic along the x and y directions, with a periodicity of approximately d along x and d/2 along y.

Finally, one can observe that in the intermediate state there is the appearance of new modes described as crosses surrounded by a circle or a square. As can be seen, their spatial profiles are impacted by the presence of edge domains. Even if the experimental intermediate state is obviously dependent on the local geometrical inhomogeneities of the antidots, we still observe a very good agreement between the experimental FMR measurements and simulated SW modes for both samples.

FIG. 3 : Micromagnetic calculations of the equilibrium magnetic states and spatial profiles of the dynamic modes for a MC with s = 200 nm and d = 800 nm. Each column shows, from the top to the bottom, the equilibrium magnetic state and the dynamic mx components calculated as described in the section II. Below each equilibrium magnetic state, a zoom around one antidot is shown as an example to observe the magnetization distribution in the edge domain. The colored symbols describe the different spin waves modes and are the same as in

B. BLS Spectra and mapping of spin

waves in quasi-saturated states.

In order to complement micromagnetic simulations and FMR experiments, twodimensional imaging of the SW modes in the quasi-saturated equilibrium state was carried out for both samples by means of micro-BLS to study the evolution of the SWs spatial profile as a function of the geometrical parameters and the intensity of the applied field.

First, micro-BLS measurements were performed measuring the SW intensity as a function of excitation frequency in the range between 2 and 16 GHz. BLS spectra recorded applying an external field 0H0 = 100 mT and 50 mT for the samples with d = 800 and 600 nm are reported in the panels (a) and (d) of the Fig. 5 and6, respectively. For both samples, the BLS spectra show two intense peaks, and a less intense peak at lower frequency.

When the external field is reduced, all the three peaks are observed to shift at lower frequencies. Note that the measured BLS frequencies of the most intense peaks agree very well with those in frequency-sweep FMR measurements (panels (b) and (e) of Fig. 5 and6) and in the calculated SW spectra (panels (c) and (f) of Fig. 5 and6).

FIG. 5 : Spin waves spectra recorded by BLS for d = 800 nm at µ0H0 = 100 mT (a) and µ0H0 = 50 mT (d). On top of each spectrum, we present the BLS intensity mapping recorded at the frequencies of the two main modes shown by the yellow and turquoise circles. Frequency-sweep FMR measurements performed for d = 800 nm at µ0H0 = 100 mT (b) and µ0H0 = 50 mT (e). FFT spectra calculated for d = 800 nm at µ0H0 = 100 mT (c) and µ0H0 = 50 mT (f). The insets in (c) and (f) present the spatial distribution of the calculated FFT amplitude at the frequencies of the two main modes shown by the yellow and turquoise circles.

In all panels, the colored symbols are the same as the ones used in the Fig. 3.

Then, for the two most intense modes, a 2D map of the BLS intensity was acquired at fixed frequency over an area of about 5 × 5 μm 2 with 250 nm step size. Such mappings are shown at the top of the panels (a) an (d) of the Fig. 5. and 6. As it can be seen, the higher frequency mode always corresponds to a localized mode, having the largest amplitude in the region comprised between the antidots, while the lower frequency mode corresponds to an extended mode propagating in the horizontal channels, perpendicularly to the external field (DE configuration), in good agreement with the calculated spatial profiles of the modes described by the yellow and turquoise circles.

Moreover, from the comparison between the experimental BLS spectra and the micromagnetic simulations, the peak at lower frequency can be labelled as an edge mode (dark blue circle). However, due to the small lateral size of the antidots, the spatial localization of this mode cannot be resolved in the two-dimensional micro-BLS maps. In agreement with previous studies, both BLS and frequency-sweep FMR measurements demonstrate that the variation of dipolar energy with d has opposite effect depending on the mode spatial localization. For the extended mode the frequency is higher in the sample with d = 800 nm while for the localized mode it is higher for d = 600 nm. As a consequence, the frequency difference between the extended and localized modes increases when decreasing d as was already clearly observed in the Fig. 2. This opposite trend as a function of the aspect ratio t/d (thickness over the width of the channels comprised between the antidots) has been ascribed to the fact that the extended modes can be considered as the resonances of transversely magnetized horizontal channels, whereas the localized mode is the lowest resonant mode of a longitudinally magnetized vertical channels [START_REF] Tacchi | Universal dependence of the spin wave band structure on the geometrical characteristics of twodimensional magnonic crystals[END_REF] . Fig. 7 shows the evolution of the BLS intensity of the extended mode recorded as a function of the distance from the antenna and for the two values of d. In order to take into account the variations of the SW propagation in the different regions of the sample, we record the signal at the center of the horizontal channels comprised between the rows of antidots and we integrate it over all horizontal channels. Finally, the intensity is normalized by its value at the first pixel position to allow comparison of decay lengths at different H0 values. The decay length D of the SW intensity can then be estimated from the fit of the intensity according to the equation:

FIG. 6 : Spin waves spectra recorded by BLS for d = 600 nm at µ0H0 = 100 mT (a) and µ0H0 = 50 mT (d). On top of each spectrum, we present the BLS intensity mapping recorded at the frequencies of the two main modes shown by the yellow and turquoise circles. Frequency-sweep FMR measurements performed for d = 600 nm at µ0H0 = 100 mT (b) and µ0H0 = 50 mT (e). FFT spectra calculated for d = 600 nm at µ0H0 = 100 mT (c) and µ0H0 = 50 mT (f). The insets in (c) and (f) present the spatial distribution of the calculated FFT amplitude at the frequencies of the two main modes shown by the yellow and turquoise circles. In all panels, the colored symbols are the same as the ones used in the

𝐼(𝑥) = exp (- 2𝑥 𝜆 𝐷 ) + 𝐼 0 ( 1
)
where x is the distance from the antenna and I0 the offset intensity.

As reported in the Fig. 7., we found that D is comprised in the range between 4 and 6.5 µm. Note that these values are of the same order of magnitude as the decay lengths observed in YIG waveguides with geometric dimensions similar to those of the channels comprised between the antidots [START_REF] Heinz | Long-range spin-wave propagation in transversely magnetized nano-scaled conduits[END_REF][START_REF]Propagation of Spin-Wave Packets in Individual Nanosized Yttrium Iron Garnet Magnonic Conduits[END_REF][START_REF] Merbouche | Frequency Filtering with a Magnonic Crystal Based on Nanometer-Thick Yttrium Iron Garnet Films[END_REF] . Moreover D values don't show a clear tendency on changing d or H0, indicating that SWs propagation is mainly affected by the local geometrical imperfections of the antidots array. Indeed, D is theoretically related to the frequency linewidth at half maximum ∆f of the resonance peak by the equation D = 2Vg/∆f, with Vg the group velocity of the SW mode. In MC, ∆f is higher than in the thin film or bulk material because of the local variation of the internal field due to the antidot shape [START_REF] Zivieri | Bragg diffraction of spin waves from a two-dimensional antidot lattice[END_REF][START_REF] Mantion | Influence of Ga+ milling on the spin waves modes in a Co2MnSi Heusler magnonic crystal[END_REF] or the SWs reflections [START_REF] Kruglyak | Damping of spin waves in a real magnonic crystal[END_REF] .

A rough estimation of Vg of this mode can still be obtained considering as a good approximation the ∆f values estimated from the frequency-sweep FMR measurements shown in the Fig. 5 (b) and 6 (b). This leads to Vg of the order of 1.1 and 0.7 km.s -1 for d = 800 and 600 nm respectively. Let's note that for d = 800 nm, while λD is different for both H0 values, Vg is roughly equivalent because the estimation of ∆f at µ0H0 = 100 mT is made difficult by the mixing of the yellow and green modes.

FIG. 7 : Evolution of spin wave intensity of the extended mode as a function of the distance x from the antenna for d = 800 nm (a) and 600 nm (b). The squares correspond to experimental data and lines correspond to the fit with the value of λD given in the

C. Spin wave modes at low fields and reconfigurable microwave spectrum at remanence.

When µ0H0 is decreased below 10 mT and down to the remanence, we observe some major differences between the SW spectra of the two samples presented in the Fig. 2. For example, the mode A and B vanish for d = 600 nm and many low intensity modes appear in the range 2-10 GHz. In opposite, for d = 800 nm, the spectrum shows similar tendencies from the intermediate state down to the remanence. Let's note that for the sake of clarity, we do not report on the Fig. 2. the frequency positions of the spin waves modes calculated at remanence. These differences are related to the cubic crystal anisotropy of the CMS which implies different equilibrium states as a function of d. For 800 nm, the crystal anisotropy allows to keep a large part of the magnetization in the horizontal channel aligned in the direction of H0, even at the remanence. As an example, the Fig. 8. (b) shows the REM Y state corresponding to the magnetic equilibrium configuration calculated after a saturation field was applied along the y direction. Because edge domains are only a little larger than in the intermediate state, the spatial profile of the 5 main calculated modes are very similar as those calculated at µ0H0 = 10 mT. This is highlighted in the Fig. 8 (b) with the colored symbols used to sort the SW modes in the Fig. 3.

On the contrary, for d = 600 nm, the cubic anisotropy is not strong enough to counter balance the antidot induced shape anisotropy. Therefore, edge domains continue to spread in the channel until they collapse, leaving magnetic domains referred as diagonal domains. Those ones connect diagonally aligned antidots and the magnetization in the domains is roughly aligned at 45° from the x or y axes as shown by the REM Y state in the Fig. 8(d). Therefore, the brutal change in the FMR spectrum observed in Fig. 2. (b) around µ0H0 ≈ 10 mT can be explained by the continuously evolving equilibrium state until the diagonal domains are formed. These ones also strongly impact the profiles of the 6 main calculated SWs modes as shown in the Fig. 8. (d).

As the cubic anisotropy impacts the CMS remanent states, it can be used to achieve reconfigurable microwave operations at zero field by stabilizing different magnetic configurations after a saturation field was applied either in the x (REM X) or y (REM y) direction. To demonstrate it, we performed for the two samples frequency-sweep FMR measurements and micromagnetic simulations in the REM Y and REM X states.

For d = 800 nm, reconfigurable SW modes are well demonstrated both in the experimental and calculated spectra presented in the Fig. 8(a). Indeed, we measure 5 main modes in the REM Y state which are strongly extinguished in the REM X state, in very good agreement with the calculated spectra. This is because most of the magnetization in the REM X state is parallel to hRF, except in the edge domains where a small part of the magnetic volume can still couple with hRF. The spatial profiles of the SWs modes in the REM X state shown in the Fig. 8(b) confirm the very weak precession amplitude for all modes except for the mode 1, for which the magnetization precession is mainly localized in the edge domains and explaining why it is not completely attenuated in the calculated spectrum.

Let's note that even if the frequencies of the calculated and experimental main SWs modes agree very well in the REM Y state, we notice that their experimental frequency linewidths are broader than in the simulation. This is attributed to the aforementioned geometrical imperfections of the antidots which lead to local variations of the equilibrium magnetic configuration. Therefore, we were not able to distinguish experimentally all the low intensity modes expected from the simulations. Also, the ∆S11 measurements presented here at remanence are exactly the same if the saturation field is 300 or 10 mT. This allows to foresee devices with a relatively small reconfigurable field which is required in the context of green technology and low energy consumption devices.

For d = 600 nm, the frequency-sweep FMR measurement presented in Fig. 8. (c) shows several well-defined SWs modes in the REM Y state confirming the small geometrical dispersion of the antidots for this sample. In opposite to the case d = 800 nm, we still observe a small signature of some of them in the REM X state. Simulations explain such features by the fact that the magnetization inside the diagonal domains can always couple to hRF. This is observed in the Fig. 8(d) where the spatial profiles of the 6 main modes show a non-negligible precession amplitude in the diagonal domain for the REM X state. As a consequence, the simulated spectra demonstrate that the modes 1, 2, 5 and 6 are not expected to be fully attenuated in the REM X state while the modes 3 and 4 and the low intensity modes in between them must be extinguished. Indeed, for these modes, the local π dephasing of the magnetization precession across the sample cancels out the averaged dynamic component. Therefore, for d = 600 nm, we can conclude that simulations describe qualitatively well the experimental spectra for the modes 2 to 6. They only fail to describe why the mode 1 is almost invisible in the frequencysweep FMR measurement in the REM X state while simulations predict only a relatively weak attenuation. For the moment we have no clear explanation for this observation. We assume that it is related to the particular spatial profile of this mode in the REM X state. Indeed, the magnetization precession is strong in the domain walls surrounding the diagonal domain. Such walls have shapes and sizes intrinsically very sensitive to even small local variations of geometry around the antidot, which are not considered in our simulations. These variations can lead to a local dephasing of the precession across the sample and then to a decrease of the signal. It is worth to mention that local variations of the shape and size of the diagonal domains or of the edge domains can also explain why the experimental signal of the mode 1 for d = 800 and 600 nm in the REM Y state is not the most intense experimental peak as expected from simulations.

To find out more about the benefits of using the cubic anisotropy to make reconfigurable microwave operations, even in the case of strongly non uniform remanent states, we compared the previous FFT spectra calculated for CMS in the REM X and REM Y states with those calculated for a MC having the same geometry but made of Permalloy. Indeed, this magnetic material is a widely used both in fundamental studies and applications. In particular, it presents a magnetization value µ0Ms = 1 T which is close to the value for the CMS (1.3 T) but it has no crystalline anisotropy. Let's note that for the sake of clarity we performed the micromagnetic simulations with a damping coefficient α = 2 x 10 -3 equal to the one used for CMS. But its normal value is about 6 x 10 -3 . The results are shown in the Fig. 9 where we also present the simulation of the remanent states in the Permalloy MCs.

For d = 800 nm, the difference with the spectra of Fig. 8(a) is obvious since most of the dynamic modes are only weakly attenuated in the REM X state. As discussed before, this is due to the presence of diagonal domains for both the REM X and REM Y states. Such domains are also present for d = 600 nm. For this d value, some differences appear with the spectra calculated for the CMS. Firstly, only one mode around 4 GHz is fully attenuated, whereas at least the modes 3 and 4 were in Fig. 8(c). Secondly, the modes at high frequencies (≥ 6 GHz) are almost not attenuated at all in the case of Permalloy. We then conclude that the cubic anisotropy of the CMS alloy favors the reconfigurability even in the case of non-uniform remanent states because it favors a more uniform magnetization in the direction of H0 in the region between the diagonal domains.

For applications, it is interesting to point out that the frequency range of the dynamic modes in the CMS (≈ 2 -10 GHz) is wider than in Permalloy (≈ 2 -8 GHz) at remanence. The possibility to have some part of the channels uniformly magnetized, such as for the sample with d = 800 nm in the present work, also favors the propagation of magnetostatic surface spin waves or magnetostatic backward volume waves for frequency filtering applications at remanence as discussed in the REF [START_REF] Mantion | Cubic Anisotropy for a Reconfigurable Magnonic Crystal Based on Co 2 Mn Si Heusler Alloy[END_REF] .

Conclusion.

In this work we performed a combined broadband FMR and micro-BLS study of SW modes in MCs made of squared antidots of lateral size 200 nm and spacing 800 or 600 nm etched in a thin film of Co2MnSi. The experimental results were explained by micromagnetic simulations, which enabled to calculate the frequency and the spatial profile of each mode, correlating their field evolution with the variation of the magnetic ground state. We demonstrated numerically and experimentally that when the intensity of the static magnetic field is >10 mT, the two main modes with the largest intensity in the spectrum of MCs correspond to a mode localized in the region between the antidots and a propagative mode extending in the channels comprised between the antidots, respectively. For both the samples the decay length of the extended mode, estimated from micro-BLS measurements, was found to be in order of few microns, comparable with the values reported for YIG waveguides having geometrical dimensions similar to those of the channels comprised between antidots. Additional modes with lower intensities are also observed, whose number depends on the amplitude of the applied static magnetic field. When the applied magnetic field is < 10 mT, the SWs spectrum of the samples undergoes a strong variation because the edge domains around the antidots spread into the channels. In particular, in the sample with d = 600 nm the edge domains are found to collapse into larger domains connecting antidots aligned diagonally.

Finally, we demonstrated that the cubic anisotropy of the Heusler alloy allows to obtain a reconfigurable microwave spectrum at remanence. Indeed, it counterbalances the local shape anisotropy induced by the antidots. For sufficiently large distance between the antidots, it favors quasi-uniform equilibrium remanent states. In particular, for a spacing of 800 nm, we achieved experimentally a very strong attenuation of SW when switching between two perpendicular quasi uniform states which are not achievable with Permalloy. Such results were very well reproduced by micromagnetic simulations. For smaller antidot spacings leading to a nonuniform equilibrium state, we have also observed reconfigurable SW modes in the CMS. Their number is higher than that calculated for a MC of the same dimensions in Permalloy, again due to the cubic anisotropy of the material.

Our study opens some perspectives for the development of reconfigurable multi-band microwave devices based on simple geometry and taking advantage of the intrinsic properties of the materials such as the crystal anisotropy, which are not usually present in soft materials as Permalloy.

FIG. 1 :

 1 FIG. 1 : (a) SEM images of a CMS squared antidot array with s = 200 nm and d = 600 nm. (b) Schematic of the unit cell used for micromagnetic simulations. 2D periodic boundary conditions are used in the x and y directions. The directions of the CMS cubic anisotropy axes are shown in orange. (c,d) Optical images of the micro-antenna's used for field-sweep or frequency-sweep FMR measurements (c) and for BLS experiments (d). The wave vector direction of the propagating SWs in BLS experiments is shown in yellow.

  Fig.2(a) and (b) present field-sweep FMR measurements for the sample with d = 800 and 600 nm, respectively. H0 is applied along the y direction, parallel to the length of the micro antenna (see Fig.1c). The intensity of the microwave absorption signal as a function of ωRF/2π and H0 is color coded in grey scale, while top panels show examples of line-cuts of the signals recorded at 15 GHz as a function of H0.Many absorption peaks, corresponding to the different SWs modes excited by the micro antenna, are observed for both samples. Moreover, the two samples exhibit a similar behavior when µ0H0 ≥ 10 mT (the case 0 ≤ µ0H0 < 10 mT will be discussed in the section III.C). In particular, two main signals, having large amplitude and referred as the mode A and B for the highest and lowest frequency peak respectively, are found in the whole investigated field range. We also observe increasing number of low frequency modes as H0 increases.These findings are very well reproduced by the micromagnetic simulations, as it can be seen in the Fig.2(a) and (b), where the simulated frequencies are reported by colored symbols for the sample with d = 800 and 600 nm, respectively. Their corresponding

Fig. 2 .

 2 FIG.4: Micromagnetic calculations of the equilibrium magnetic states and spatial profiles of the dynamic modes for a MC with s = 200 nm and d = 600 nm. Each column shows, from the top to the bottom, the equilibrium magnetic state and the dynamic mx components calculated as described in the section II. Below each equilibrium magnetic state, a zoom around one antidot is shown as an example to observe the magnetization distribution in

Fig. 4 .

 4 FIG. 6 : Spin waves spectra recorded by BLS for d = 600 nm at µ0H0 = 100 mT (a) and µ0H0 = 50 mT (d). On top of each spectrum, we present the BLS intensity mapping recorded at the frequencies of the two main modes shown by the yellow and turquoise circles. Frequency-sweep FMR measurements performed for d = 600 nm at µ0H0 = 100 mT (b) and µ0H0 = 50 mT (e). FFT spectra calculated for d = 600 nm at µ0H0 = 100 mT (c) and µ0H0 = 50 mT (f). The insets in (c) and (f) present the spatial distribution of the calculated FFT amplitude at the frequencies of the two main modes shown by the yellow and turquoise circles. In all panels, the colored symbols are the same as the ones used in the Fig. 4.

FIG. 8 :

 8 FIG. 8 : Frequency-sweep FMR measurement (top) and calculated FFT spectra (bottom) for the REM Y and REM X remanent states when d = 800 nm (a) and 600 nm (c). Simulations for d = 800 nm (b) and 600 nm (d) of the equilibrium REM Y and REM X states and of the mx spatial profile of the main dynamic modes numbered in (a) and (c) respectively.

FIG. 9 :

 9 FIG. 9 : FFT spectra in the REM Y and REM X remanent states calculated for a MC made of Permalloy with squared antidots with s = 200 nm and d = 800 nm (a) or 600 nm (b). Below each spectrum is shown the corresponding REM X and REM Y states.

Acknowledgement.

This study has been partially supported through the French National Research Agency for financial support within the ANR program EHIS (ANR-21-CE42-0003) and the EUR grant NanoX n° ANR-17-EURE-0009 in the framework of the « Programme des Investissements d'Avenir". This work was performed using HPC resources from CALMIP (Grant 2023-p1554). This work has been funded by the European Union -NextGenerationEU under the Italian Ministry of University and Research (MUR) National Innovation Ecosystem grant ECS00000041 -VITALITY. CUP: B43C22000470005; J97G22000170005.

Data availability.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Author declarations

Conflict of Interest


Gautami Bhowmik

Jan-Christoph Schlage-Puchta

Essential singularities of Euler products

published or not. The documents may come

Introduction and results

M any D i ri chl et-seri esoccurri ng i n practi ce sati sfy an Eul er-product,and i fthey do so,the Eul er-product i s often the easi est way to access the seri es. T herefore, i t i s i m portant to deduce i nform ati on on the seri es from the Eul er-product representati on. O ne ofthe m ost i m portant appl i cati ons ofD i ri chl et-seri es,goi ng back to R i em ann,i sthe asym ptoti c esti m ati on ofthe sum ofi ts coe ci entsvi a Perron' s form ul a,that i s,the use ofthe equati on

X n x a n = 1 2 i c+ i1 Z c i1 X n 1 a n n s
x s s ds:

To use thi s rel ati on,one usual l y shi fts the path ofi ntegrati on to the l eft,thereby reduci ng the contri buti on ofthe term x s .T hi sbecom espossi bl e onl y i fthe functi on D (s)= P an n s i shol om orphi c on the new path and therefore the questi on ofconti nuati on ofD i ri chl et-seri es beyond thei r dom ai n ofabsol ute convergence i s a central i ssue i n thi s theory.In fact,the i m portance ofthe R i em ann hypothesi sstem s from the fact that i t woul d al l ow us to m ove the path ofi ntegrati on for D (s)= 0 (s) to the l i ne 1=2 + w i thout m eeti ng any si ngul ari ty besi des the obvi ous pol e at 1.

Esterm ann [START_REF] Sterm Ann | O n certain functions represented by D irichletseries[END_REF]appears to be the rst to address thi s probl em . H e showed that for an i nteger val ued pol ynom i alW (x) w i th W (0)= 1 the D i ri chl et-seri es D (s)= Q p W (p s ) can ei ther be w ri tten as a ni te product of the form Q N ( s) c for certai n i ntegers c ,and i s therefore m erom orphi cal l y conti nuabl e to the w hol e com pl ex pl ane,or i s conti nuabl e to the hal f-pl ane < s > 0. In the l atter case the l i ne < s = 0 i s the naturalboundary of the D i ri chl et-seri es. T he strategy of hi s proofwas to show that every poi nt on the l i ne < s = 0 i s an accum ul ati on poi nt of pol es or zeros of D . N ote that , the R i em ann-zeta functi on i tsel f, does not fal lam ong the cases under consi derati on,si nce W (X ) = (1 X ) 1 i s a rati onal functi on. D ahl qui st [START_REF] Ahlquist | O n the analytic continuation of E ulerian products[END_REF]general i zed Esterm ann' s work al l ow i ng W to be a functi on hol om orphi c i n the uni t ci rcl e w i th the excepti on of i sol ated si ngul ari ti es and i n parti cul arcoveri ng thecasethatW be rati onal .T hi sm ethod ofproofwasextended to m uch greatergeneral i ty,i nterestbei ng sparked by -functi onsofni l potentgroups i ntroduced by G runewal d,Segaland Sm i th [START_REF] Segal | Subgroups of nite index in nilpotent groups[END_REF]as wel las hei ght zeta functi ons [START_REF]Sw ynnerton-D yer, Fonction zêta des hauteurs associ ee a une certaine surface cubique[END_REF] . Functi ons ari si ng i n these contexts are often of the form D (s) = Q W (p;p s ) for an i ntegralpol ynom i alW . D u Sautoy and G runewal d [START_REF] Sautoy | Zeta functions of groups: zeros and friendly ghosts,A m er[END_REF]gave a cri teri on for such a functi on to have a naturalboundary w hi ch,i n a probabi l i sti c sense,appl i es to al m ost al lpol ynom i al s. A gai n, i t i s show n that every poi nt on the presum ed boundary i s an accum ul ati on poi nt ofzeros or pol es. T he fol l ow i ng conjecture,see for exam pl e [ 8,1.11][ 7,1.4],i s bel i eved to be true.

C onjecture 1. Let W (x;y) =

P

n ;m a n ;m x n y m be an integral pol ynom ial with W (x;0) = 1. T hen D (s) = Q p W (p;p s ) is m erom orphicall y continuabl e to the whol e com pl ex pl ane ifand ifonl y ifitis a nite product ofR iem ann -functions. M oreover,in the l atter case if = m axf n m :m 1;a n ;m 6 = 0g,then < s = is the naturalboundary ofD .

In thi s paper we show that any re nem ent ofEsterm ann' s m ethod i s bound to fai lto prove thi s conjecture.

IfW (X ;Y ) i s a rati onalfuncti on,we expand W i nto a power seri es W (X ;Y )= P n ;m 0 a n ;m X n Y m ,and de ne = supf n + 1 m :m 1;a n ;m 6 = 0g, = supf n m : m 1;a n ;m 6 = 0g. It i s easy to see that the suprem um i s actual l y attai ned,and that the functi on W = 1 + P n m = a n ;m X n Y m i s agai n a rati onalfuncti on. W e cal l W the m ai n part ofW ,si nce onl y W i s responsi bl e for the convergence ofthe productD (s). ForW a pol ynom i al W wascal l ed the ghostofW i n [START_REF] Sautoy | Zeta functions of groups: zeros and friendly ghosts,A m er[END_REF] . A rati onal functi on W i s cal l ed cycl otom i c i fi t can be w ri tten as the product ofcycl otom i c pol ynom i al s and thei r i nverses.

W e de ne an obstructi ng poi nt z to be a com pl ex num ber w i th < z = ,such that there exi sts a sequence ofcom pl ex num bers z i ,< z i > ,z i ! z,such that D hasa pol e ora zero i n z i for al li. O bvi ousl y,each obstructi ng poi nt i s an essenti al si ngul ari ty for D ,the converse not bei ng true i n general .

O ur m ai n resul t i s the fol l ow i ng.

T heorem 1. LetW (X ;Y ) be a rationalfunction,which can be written as P (X ;Y ) Q (X ;Y ) , where P;Q 2 Z[ X ;Y ]satisfy P (X ;0)= Q (X ;0)= 1. D e ne a n ;m ; ; W and D as above. T hen the product representation ofD converges in the hal f-pl ane < s > , D can m erom orphicall y continued into the hal f-pl ane < s > ,and precisel y one of the foll owing hol ds true.

(1) W is cycl otom ic and once its unitary factors are rem oved,W = W ;in this case D is a nite productofR iem ann -functions; (2) W is not cycl otom ic; in this case every point of the l ine < s = is an obstructing point;

(3) W 6 = W , W is cycl otom ic and there are in nitel y m any pairs n;m with a n ;m 6 = 0 and n m < < n + 1 m ;in this case is an obstructing point; (4) W 6 = W , W is cycl otom ic, there are onl y nitel y m any pairs n;m with a n ;m 6 = 0 and n m < < n + 1 m , but there are in nitel y m any prim es p such that the equation W (p;p s ) = 0 has a sol ution s 0 with < s 0 > ; in this case every pointofthe l ine < s = is an obstructing point;

(5) N one ofthe above;in thiscase no pointon the l ine < s = isan obstructing point.

W e rem ark that each of these cases actual l y occurs, that i s, there are Eul erproducts for w hi ch Esterm ann' s approach cannot work. N oti ce that w hi l e i n the thi rd case we need i nform ati on on the zeros of the R i em ann-zeta functi on to know about the m erom orphi c conti nuati on,i n the l ast case we can say nothi ng about thei r conti nuati on.

W hi l e the above cl assi cati on l ookspretty techni cal ,these casesactual l y behave qui te di erentl y. To i l l ustrate thi s poi nt we consi der a dom ai n C w i th a functi on f : ! C ,l et N ( ) the num ber ofzeros and pol es off i n counted w i th posi ti ve m ul ti pl i ci ty,that i s,an n-fol d zero or a pol e oforder n i s counted n ti m es. T hen we have the fol l ow i ng.

C orollary 1. Let W be a rationalfunction, and de ne as above. T hen one of the foll owing two statem ents hol ds true:

(1) For every > 0 we have

N (fj z j< ;< z > 0g)= 1 ; (2) W e have N (f< z > ;j = zj< T g = O (T l ogT ).
If W isa pol ynom ialand we assum e the R iem ann hypothesis aswellasthe Q -l inear independence of the im aginary parts ofthe non-trivialzeros of , then there exist constants c 1 ;c 2 ,such thatN

(f< z > ;j = zj< T g)= c 1 T l ogT + c 2 T + O (l ogT ).
Fi nal l y we rem ark that for -functi ons ofni l potent groups the general i zati on to rati onalfuncti onsi si rrel evant,si nce a resul tofdu Sautoy [START_REF] Du Sautoy | Zeta functions ofgroups and rings: uniform ity[END_REF]

i m pl i esthati f G (s)= Q p W (p;p s ) for a rati onalfuncti on W (X ;Y ) = P (X ;Y ) Q (X ;Y )
,then Q i s a cycl otom i c pol ynom i al ,that i s, G can be w ri tten as the product of ni tel y m any R i em ann -functi ons and a D i ri chl et-seri es ofthe form

Q p W (p;p s ) w i th W a pol ynom i al .
H owever,for other appl i cati ons i t i s i ndeed i m portant to study rati onalfuncti ons, one such exam pl e occurs i n the recent work of de l a B ret eche and Sw ynnerton-D yer [START_REF]Sw ynnerton-D yer, Fonction zêta des hauteurs associ ee a une certaine surface cubique[END_REF] .

P roof of case 2

In thi s secti on we show that i f W i s not cycl otom i c,then < s = i s the naturalboundary of the m erom orphi c conti nuati on of D . For W a pol ynom i althi s was show n by du Sautoy and G runewal d [START_REF] Sautoy | Zeta functions of groups: zeros and friendly ghosts,A m er[END_REF] ,our proofcl osel y fol l ow s thei r l i nes of reasoni ng.

T he m ai n di erence between the case ofa pol ynom i aland a rati onalfuncti on i s that for pol ynom i al s the l ocalzeros created by di erent pri m es can never cancel , w hereasfora rati onalfuncti on the zerosofthe num eratorbel ongi ng to som e pri m e num ber p m i ght coi nci de w i th zeros ofthe denom i nator bel ongi ng to som e other pri m eq,and m ay thereforenotcontri buteto the zerosorpol esneeded to provethat som e poi nton the presum ed boundary i sa cl usterpoi nt.W e coul d excl ude the possi bi l i ty ofcancel l ati onsby assum i ng som e unproven hypothesesfrom transcendence theory,however,here we show that we can dealw i th thi s case uncondi ti onal l y by provi ng thatthe am ountofcancel l ati on rem ai nsl i m i ted. W e rstconsi derthe case of cancel l ati ons between the num erator and denom i nator com i ng from the sam e pri m e num ber.

Lem m a 1. LetP;Q 2 Z[ X ;Y ]be co-prim e non-constantpol ynom ial s. T hen there are onl y nitel y m any prim es p, such that for som e com pl ex num ber s we have P (p;p s )= Q (p;p s )= 0.

Proof. Let V be the vari ety of hP;Q i over C . A ssum e there are i n ni tel y m any pai rs (p;s),for w hi ch the equati on P (p;p s )= Q (p;p s )= 0 hol ds true. T hen V i s i n ni te, hence,at l east one-di m ensi onal . Si nce P and Q are non-constant,we have V 6 = C 2 ,hence,V i sone-di m ensi onal .LetV 0 be a one-di m ensi onali rreduci bl e com ponent, and l et R be a generator of the i deal correspondi ng to V 0 . T hen hP;Q i hR i,that i s,R di vi des P and Q ,w hi ch i m pl i es that R i s constant. B ut a constantpol ynom i alcannotde ne a one-di m ensi onalvari ety and thi scontradi cti on provesour cl ai m .

N ext we use the fol l ow i ng graph-theoreti c resul t, descri bi ng graphs w hi ch are rather cl ose to trees. W e cal la cycl e i n a graph m inim ali fi t i s ofl ength 3,and not the uni on oftwo cycl es ofsm al l er l ength.

Lem m a 2. Let G be a graph, k 2 an integer,such that every vertex has degree 3k,and thatthere exists a sym m etric rel ation on the vertices,such thatevery vertex v is in rel ation to atm ostk other vertices,and every m inim alcycl e passing through v al so passes through one ofthe vertices in rel ation to v. T hen G is in nite.

Proof. Suppose that G were ni te, and x som e vertex v 0 . W e cal l a geodesi c path good i fno two verti cesofthe path stand i n rel ati on to each other.W e wantto constructan i n ni te good path.N ote thatp 1 and p 2 aregood pathsof ni te l ength, they cannot i ntersect i n but one poi nt,for otherw i se thei r uni on woul d contai n a cycl e,and choosi ng one ofthe i ntersecti on poi nts we woul d obtai n a contradi cti on w i th the de ni ti on ofa good path. H ence, the uni on of the good paths starti ng i n v 0 form s a tree. T here are 3k verti ces connected to v 0 ,at m ost k ofw hi ch are forbi dden. H ence,the rst l ayer ofthe tree contai ns at l east 2k poi nts. Each ofthese poi nts i s connected to at l east 3k other poi nts. It stands i n rel ati on to at m ostk ofthem and hence we can extend every path i n at l east2k ways,and ofal l these paths at m ost k stand i n rel ati on w i th v 0 . H ence,the second l ayer contai ns at l east 4k k poi nts. D enote by n i the num ber ofpoi nts i n the i-th l ayer ofthe tree. T hen,conti nui ng i n thi s way,we obtai n

n i+ 1 2kn i k(n i 1 +
+ n 1 + 1): From thi s and the assum pti on that k 2 i t fol l ow s by i nducti on that n i+ 1 kn i , hence,the tree and therefore the graph G,w hi ch contai ns the tree,i s i n ni te.

N otethei m portanceofsym m etry:i ftherel ati on i sal l owed to benon-sym m etri c, we can get two regul ar trees,and i denti fy thei r l eaves. T hen every m i ni m alcycl e passi ng through one poi nt ei ther passes through i ts parent node or the m i rror i m age ofthe poi nt. T husi n the absence ofsym m etry the resul tbecom esw rong for arbi trari l y l arge val ency even for k = 2.

W e can now prove our resul t on non-cancel l ati on.

Lem m a 3. Let P;Q 2 Z[ X ;Y ]be co-prim e pol ynom ial s with de ned as in the introduction. Let > 0 be given,and suppose that for a prim e p 0 su cientl y l arge P (p 0 ;p s 0 ) has a zero on the segm ent [ + it; + it+ ] , where > . T hen

Q p P (p;p s ) Q (p;p s )
has a zero or a pol e on this segm ent. Proof. Si nce the l ocalzeros converge to the l i ne < s = , there are onl y ni tel y m any pri m es p forw hi ch the num eratoror denom i natorhas a zero,hence,we m ay assum e that P (p;p s );Q (p;p s )6 = 0 for p > p 0 . For each pri m e p l et z p 1 ;:::;z p k be the roots ofthe equati on P (p;p s ) = 0 i n the segm ent < s = ,0 = s 2 log p , and l etw p 1 ;:::;w p ' be the rootsofthe equati on Q (p;p s )= 0 on thi ssegm ent.Such roots need not exi st but i fthey do then thei r num ber i s bounded i ndependentl y of p. T he rootsofthe equati onsP (p;p s )= 0 and Q (p;p s )= 0 form a pattern w i th peri od 2 i log p . Ifp 0 i s su ci entl y l arge,then becom esarbi trary sm al l ,hence,i fp i s notl arge then the equati onsP (p;p s )= 0 and Q (p;p s )= 0 do nothave sol uti ons on the l i ne < s = + . Let p 1 be the l east pri m e for w hi ch such sol uti ons exi st. For p 1 su ci entl y l arge and p > p 1 ,ei ther P (p;p s ) = 0 has no sol uti on on the segm entunderconsi derati on ori thasatl east log p 2 such sol uti ons.N ote thatby xi ng and choosi ng p 0 su ci entl y l argewe can m akethi sexpressi on asl argeaswe need. Further note that by choosi ng p 0 l arge we can ensure,i n vi ew ofLem m a 1, that P (p;p s )= Q (p;p s )= 0 has no sol uti on on the l i ne < s = + .

W e now de ne a bi parti te graph G as fol l ow s: T he verti ces ofthe graph are al l com pl ex num bersz p i i n one setand al lcom pl ex num bersw p i i n the otherset,w here p p 0 . T wo verti cesz p i ,w q j are joi ned by an edge i fthere exi stsa com pl ex num ber s w i th < s = + ,t = s t+ ,such thats i scongruentto z p i m odul o 2 i log p ,and congruent to w q j m odul o 2 i log q . In other words,the exi stence ofan edge i ndi cates thatone ofthe zerosofP (p;p s ) obtai ned from z p i by peri odi ci ty cancel sw i th one zero ofQ (q;q s )obtai ned from w q j . If

Q p P (p;p s ) Q (p;p s
) hasnei ther a zero nora pol e on the segm ent,then every zero ofone ofthe pol ynom i al s cancel s w i th a zero ofthe other pol ynom i al ,that i s,every vertex has val ency at l east log p1 for som e i 6 = j. T he rel ati on de ned by x p i x q j , p = q, x 2 fz;w g i s an equi val ence rel ati on w i th equi val ence cl asses bounded by som e constant K . Ifwe choose p 1 > exp(6 K 1 ), the assum pti ons of Lem m a 2 are sati s ed, and we concl ude that G i s ni te.

B ut we al ready know that there i s no p > p 0 for w hi ch P (p;p s ) = 0 or Q (p;p s ) = 0 has a sol uti on, that i s, G i s ni te. T hi s contradi cti on com pl etes the proof. U si ng Lem m a 3 the proofnow proceeds i n the sam e fashi on as i n the pol ynom i alcase;for the detai l s we refer the reader to the proofgi ven by du Sautoy and G runewal d [START_REF] Sautoy | Zeta functions of groups: zeros and friendly ghosts,A m er[END_REF] .

D evelopm ent in cyclotom ic factors

A rati onalfuncti on W (X ;Y ) w i th W (X ;0) = 1 can be w ri tten as an i n ni te product ofpol ynom i al s ofthe form (1 X a Y b ). H ere convergence i s m eant w i th respectto the topol ogy ofform alpowerseri es,thati s,a product

Q 1 i= 1 (1 X ai Y bi ) convergesto a powerseri esf i fforeach N thereexi stsan i 0 ,such thatfori 1 > i 0 the parti alproduct Q i1 i= 1 (1 X ai Y bi
)coi nci desw i th f foral lcoe ci entsofm onom i al s X a Y b w i th a;b < N . T he exi stence ofsuch an extensi on i squi te obvi ous,however, we need som e expl i ci t i nform ati on on the factors that occur and we shal ldevel op the necessary i nform ati on here.

For a set A R 2 de ne the convex cone A generated by A to be the sm al l est convex subset contai ni ng a for al la 2 A and > 1. A poi nt a ofA i s extrem al , i fi t i s contai ned i n the boundary ofA and there exi sts a tangentto A i ntersecti ng A preci sel y i n a,or set theoreti cal l y speaki ng,i fA n fag 6 = A . N ote that a convex cone form s an addi ti ve sem i -group as a subsem i group ofR 2 .

To a form alpowerseri esW =

P n ;m a n ;m X n Y m 2 Z[ [ X ;Y ] ]we associ ate the set A W = f(n;m ):a n ;m 6 = 0g.Suppose westartw i th a rati onalfuncti on W 2 Z[ X ;Y ] , w hi ch i s ofthe form P (X ;Y ) Q (X ;Y ) w i th P (X ;0)= Q (X ;0)= 1. T hen 1 Q (X ;Y ) = 1 X = 0 Q (X ;Y ) 1 = X n ;m b n ;m X n Y m ;
say,w here the convergence ofthe geom etri c seri es as a form alpower seri es fol l ow s from the factthat every m onom i ali n Q i s di vi si bl e by Y . T he setf(n;m ):b n ;m 6 = 0g R 2 i s contai ned i n the sem i group generated by the poi nts correspondi ng to m onom i al s i n Q ,but m ay be stri ctl y sm al l er,as there coul d be unforeseen cancell ati ons. M ul ti pl yi ng the power seri es by P (X ;Y ),we obtai n that A W i s contai ned w i thi n ni tel y m any shi fted copi es ofA Q 1 . Let(n;m )bean extrem alpoi ntofA w .T hen wehaveW = (1

X n Y m ) an ;m W 1 (X ;Y ), w here W 1 (X ;Y ) = (1 X n Y m ) an ;m W (X ;Y ). O bvi ousl y, W 1 (X ;Y ) i s a form al powerseri esw i th i ntegercoe ci ents,we cl ai m that A W 1 i sa propersubsetofA W .
In fact,the m onom i al s ofW 1 are obtai ned by taki ng the m onom i al s ofA W ,m ul tipl yi ng them by som e power ofX n Y m ,and possi bl y addi ng up the contri buti on of di erent m onom i al s. H ence,A W 1 i s contai ned i n the sem i group generated by A W . B ut (n;m ) i s noti n A W 1 ,and si nce (n;m ) was assum ed to be extrem al ,we obtai n

A W 1 hA W in f(n;m )g A W n f(n;m )g A W :
Taki ng the convex cone i s a hul l operator, thus A W 1 i s a proper subset of A W . Si nce we begi n and end w i th a subset ofN 2 ,we can repeat thi s procedure so that after ni tel y m any steps the resul ti ng power seri es W k contai ns no non-vani shi ng coe ci ents a n ;m w i th n < N ;m < M . T hi s su ces to prove the exi stence of a productdecom posi ti on,i n fact,i fone i s not i nterested i n the occurri ng cycl otom i c factors one coul d avoi d power seri es and stay w i thi n the real m ofpol ynom i al s by setti ng W 1 (X ;Y ) = (1 + X n Y m ) an ;m W (X ;Y ) w henever a n ;m i s negati ve. H owever,i n thi s way we trade one operati on i nvol vi ng power seri es for i n ni tel y m any i nvol vi ng pol ynom i al s,w hi ch i s better avoi ded for actualcal cul ati ons.

W hi l e we can easi l y determ i ne a super-set ofA W 1 ,i n generalwe cannot prove thatsom e coe ci entofW 1 does notvani sh,thati s,know i ng onl y A W and notthe coe ci ents we cannot show that A W 1 i s as l arge as we suspect i t to be. H owever, i ti seasy to see thatw hen el i m i nati ng one extrem alpoi ntal lotherextrem alpoi nts rem ai n untouched. In parti cul ar, i f we want to expand a pol ynom i al W i nto a product ofcycl otom i c pol ynom i al s,at som e stage we have to use every extrem al poi ntofA W ,and the coe ci entattached to thi s poi nthasnotchanged before thi s step,by i nducti on i t fol l ow s that the expansi on as a cycl otom i c product i s uni que.

W e now assum e that W i s cycl otom i c,w hi l e W i s not. W e further assum e that W i s a pol ynom i al ,and that the num erator ofW i s not di vi si bl e by a cycl otom i c pol ynom i al . W e can al ways sati sfy these assum pti ons by m ul ti pl yi ng or di vi di ng W w i th cycl otom i c pol ynom i al s,w hi ch corresponds to the m ul ti pl yi ng or di vi di ng D w i th certai n shi fted -functi ons,and does not change our probl em . O ur ai m i s to nd som e i nform ati on on the setf(n;m ):c n ;m 6 = 0g,w here the coe ci entsc n ;m are de ned vi a the expansi on W (X ;Y )=

Q

(1 X n Y m ) cn ;m . In the rststep werem oveal lpoi ntson thel i ne n m = .B y assum pti on wecan do so by usi ng ni tel y m any cycl otom i cpol ynom i al s.T heresul ti ng powerseri esbeW 1 . T he i nverseofthe productof ni tel y m any cycl otom i cpol ynom i al si sa powerseri es w i th pol esatcertai n rootsofuni ty,hence,wecan expressthesequenceofcoe ci ents as a pol ynom i al i n n and R am anujan-sum s c d (n) for d di vi di ng som e i nteger q. C onsi der som e poi nt (n;m ) 2 A W , and com pute the coe ci ent attached to thi s poi nti n W 1 . IfA W does notcontai n a poi nt(n 0 ;m 0 ),such that (n n 0 ;m m 0 ) i s col l i nearto ( ;1),then thi scoe ci enti scl earl y 0.O therw i se we consi deral lpoi nts (n 1 ;m 1 );:::;(n k ;m k ) i n A W ,w hi ch are on the paral l elto ( t;t) through (n;m ). T he coe ci ents of W 1 attached to poi nts on ' are l i near com bi nati ons of shi fted coe ci ents ofi nverse cycl otom i c pol ynom i al s,hence,they can be w ri tten as som e pol ynom i alw i th peri odi c coe ci ents. In parti cul ar,ei ther there are onl y ni tel y m any non-vani shi ng coe ci ents,or there exi sts a com pl ete ari thm eti c progressi on ofnon-vani shi ng coe ci ents. H ence,we nd thatA W 1 i scontai ned w i thi n a l ocal l y ni te setofl i nesparal l elto ( t;t),and every l i ne ei thercontai nsonl y ni tel y m any poi nts,ora com pl ete ari thm eti c progressi on.Suppose thatevery l i ne contai nsonl y ni tel y m any poi nts. T hen there exi sts som e 0 > ,such that A W 1 i s contai ned i n f(s;t) :s 1 tg,i n parti cul ar,W 1 i s regul ar i n f(z 1 ;z 2 ) :j z 1 j j z 2 j g. B ut W 1 = P Q W ,and by assum pti on P i snotdi vi si bl e by W ,therefore,there exi stpoi nts (z 1 ;z 2 ), w here W vani shes,but P does not,and these poi nts are si ngul ari ti es of W 1 sati sfyi ng j z 1 j j z 2 j . H ence, there exi sts som e l i ne contai ni ng a com pl ete ari thm eti c progressi on.

Let (x;0) + t( ;1) be the uni que l i ne contai ni ng i n ni tel y m any el em ents of A W 1 ,such that for each y > 0 the l i ne (y;0)+ t( ;1) contai ns onl y ni tel y m any el em ents (n 1 ;m 1 );:::;(n k ;m k ) of A W 1 . Set = x,that i s,the di stance ofthe ri ght boundary ofA W from the l i ne (x;0)+ t( ;1) m easured hori zontal y,and set i = m i =n i ,that i s, i i s the di stance of(n i ;m i ) from the ri ght boundary,al so m easured hori zontal l y,and = m i n i > 0. W e now el i m i nate the poi ntsa i to obtai n the powerseri esW 2 .W hen doi ng so we i ntroduce l otsofnew el em entsto the l eftofthe l i ne (x;0)+ t( ;1),w hi ch are ofno i nterestto us,and ni tel y m any poi ntson thi sl i neorto theri ghtofthi sl i ne,i n fact, we can getpoi ntsatm ostatthe poi ntsofthe form (n i ;m i )+ (n j ;m j ), ; 2 N , ; > 0.N ote thatthe hori zontaldi stance from the l i ne t( ;1)i saddi ti ve,thati s, A W 2 i scontai ned i n the i ntersecti on ofA W and the hal f-pl ane to the l eftofthe l i ne (x;0)+ t( ;1),together w i th ni tel y m any poi nts between the l i nes (x;0)+ t( ;1) and t( ;1),each ofw hi ch has di stance atl east2 from the l atter l i ne. R epeati ng thi sprocedure,we can agai n doubl e thi sdi stance,and after ni tel y m any stepsthi s m i ni m aldi stance i s l arger than the w i dth ofthe stri p,w hi ch m eans that we have arri ved at a power seri es W 3 such that A W 3 i s contai ned i n the i ntersecti on ofA W and the hal f-pl ane to the l eft ofthe l i ne (x;0)+ t( ;1). M oreover,si nce at each step there are onl y ni tel y m any poi nts changed on the l i ne (x;0)+ t( ;1) , we see that the i ntersecti on ofA W 3 w i th thi s l i ne equal s the i ntersecti on ofA W 1 w i th thi s l i ne up to ni tel y m any i ncl usi ons or om i ssi ons. Si nce an i n ni te ari thm eti c progressi on,from w hi ch ni tel y m any poi nts are del eted sti l lcontai ns an i n ni te ari thm eti c progressi on,we see thatA W 3 contai nsan i n ni te ari thm eti c progressi on.

N extweel i m i natethepoi ntson A W starti ng atthebottom and worki ng upwards. W hen el i m i nati ng a poi nt,we i ntroduce (possi bl y i n ni tel y m any)new poi nts,but al lofthem are on the l eft ofthe l i ne (x;0)+ t( ;1). H ence,after i n ni tel y m any stepswe arri ve ata powerseri esW 4 ,forw hi ch A W 4 i scontai ned i n the i ntersecti on ofA W and the open hal f-pl ane to the l eft of(x;0)+ t( ;1).

Fortunatel y, from thi s poi nt on we can be l ess expl i ci t. C onsi der the set of di erences ofthe sets A W i from the l i ne t( ;1). Taki ng the di erences i s a sem igroup hom om orphi sm , hence, at each stage the set of di erences i s contai ned i n the sem i -group generated by the di erences we started w i th. B ut si nce W i s a pol ynom i al , thi s sem i -group i s ni tel y generated, and therefore di screte. H ence, no m atter how we el i m i nate term s, at each stage the set A W i i s contai ned i n a set ofparal l el s to t( ;1) i ntersecti ng the realaxi s i n a di screte set ofnon-posi ti ve num bers.

C ol l ecti ng the cycl otom i c factorsused duri ng thi sprocedure,we have proven the fol l ow i ng.

Lem m a 4. LetW (X ;Y ) be a rationalfunction such that W (X ;Y ) is a cycl otom ic pol ynom ial , but W itsel f is not cycl otom ic. D e ne as above. T hen there is a unique expansion W (X ;Y )=

Q n ;m (1 X n Y m ) cn ;m .
T he setC = f(n;m ):c n ;m 6 = 0g contains an in nite arithm etic progression with di erence a m ul tipl e of ( ;1), onl y nitel y m any el em ents to the right of this l ine, and all entries are on l ines parall el to t( ;1), such that the l ines intersect the real axis in a discrete set of points.

P roof of case 3

W e prove that i s an obstructi ng poi nt. For i ntegers n;m w i th c n ;m 6 = 0 the factor ( n + m s) cn ;m createsa pol e or a zero at n + 1 m ,w hi ch for n + 1 m > i s to the ri ght ofthe supposed boundary. H ence,i f i s not an obstructi ng poi nt,for som e > 0 and al lrati onalnum bers 2 ( ; + ) we woul d have

P n + 1 m
= c n ;m = 0. W e now show that thi s i s i m possi bl e by provi ng that there are pai rs (n;m ) w i th n + 1 m arbi trari l y cl ose to ,c n ;m 6 = 0,such that the sum consi sts ofa si ngl e term ,and i s therefore non-zero as wel l .

Let k ' be the sl ope ofthe rays. Let f(n i ;m i )g be a l i st ofthe starti ng poi nts of the raysdescri bed i n Lem m a 4,w here (n 0 ;m 0 )de nesthe ri ght-m ostray.Take an i nteger q,such that c km + n0 ;'q + m 0 6 = 0 for al lbut ni tel y m any naturalnum bers . Let d be the greatestcom m on di vi sor ofm 0 and q. T he pri m e num ber theorem for ari thm eti c progressi ons guarantees i n ni tel y m any ,such that ' + m 0 d = p i s pri m e.Suppose there i sa pai rn 0 ;m 0 bel ongi ng to anotherray,such thatc n 0 ;m 0 6 = 0 and n 0 + 1 m 0 = k + n0 ' + m 0 . T he poi nt (n 0 ;m 0 ) m ust l i e on one ofthe ni tel y m any rays, hence, we can w ri te n 0 = k 0 + n 1 , m 0 = ' 0 + m 1 . Si nce p i s a di vi sor of the denom i nator of the ri ght hand si de,i t al so has to di vi de the denom i nator ofthe l eft hand si de. W e obtai n that p di vi des both ' + m 0 and ' 0 + m 1 . R estri cti ng, i fnecessary,to an ari thm eti c progressi on,we obtai n an i n ni tude ofi ndi ces such that' 0 + m 1 = t(' + m 0 ),w here t2 [ 0;1]i s a rati onalnum berw i th denom i nator di vi di ng d. H ence,we obtai n that the equati ons

' 0 + m 1 = t(' + m 0 ); t(k 0 + n 1 )= k + n 0
have i n ni tel y m any sol uti ons ; 0 2 N . T wo l i near equati ons i n two vari abl es, none ofw hi ch i s tri vi al ,can onl y have i n ni tel y m any sol uti ons,i fthese equati ons are equi val ent,that i s,t 2 = 1,w hi ch i m pl i es t= 1 si nce t i s posi ti ve by de ni ti on. H ence,w ri ti ng the equati ons as vectors,we have

( 0 ) k ' = n 1 m 1 n 0 m 0 ;
that i s, the vector l i nki ng n 0 m 0 w i th n 1 m 1 i s col l i near w i th k ' , contrary to the assum pti on that n 0 ;m 0 was on a ray other than that ofn;m . H ence,pol es offactors accum ul ate at . It rem ai ns to check that these pol es are not cancel l ed by zerosofotherfactors.Si nce zerosof -factorsare neverposi ti ve real s,these factors do not cause probl em s. Suppose that a pol e of (ns m ) cancel s w i th a zero of the l ocalfactor W (p;p s ),that i s,W (p;p (m + 1)=n )= 0. Si nce W has coe ci ents i n Z,thi s i m pl i es that p (m + 1)=n i s al gebrai c ofdegree atm ostequalto the degree ofW ,hence, m + 1 n can be reduced to a fracti on w i th denom i nator at m ost equal to the degree ofW . T here are onl y ni tel y m any rati onalnum bers i n the i nterval [ ; + 1]w i th bounded denom i nator,hence,onl y ni tel y m any ofthe pol es can be cancel l ed,that i s, i s i n fact an obstructi ng poi nt.

For the corol l ary note that i n cases (2){( 4) i s an obstructi ng poi nt,that i s,i n these cases the rst condi ti on ofthe corol l ary hol ds true. In case (1) and ( 5),we can representD asthe productof ni tel y m any R i em ann -functi onsm ul ti pl i ed by som e functi on w hi ch i s hol om orphi c i n the hal f-pl ane < s > ,and has zeros onl y w here the ni tel y m any l ocalfactors vani sh. A l ocalfactor bel ongi ng to the pri m e p creates a 2 i log p -peri odi c pattern of zeros, hence, the num ber of zeros and pol es i s bounded above by the num ber ofzeros ofthe ni tel y m any -functi ons,w hi ch i s O (T l ogT ),and the ni tel y m any sets ofperi odi c patterns,w hi ch create O (T ) zeros.H ence,N (f< z > ;j = zj< T g)i sO (T l ogT ).Itm ay happen thatthere are si gni cantl y l esspol esorzeros,i fpol es ofone factorcoi nci de w i th pol esofanother factor,however,wecl ai m thatunderR H and theassum pti on ofl i neari ndependence ofzerosthe am ountofcancel l ati on i snegl i gi bl e.Fi rst,i fthe i m agi nary partofzeros of areQ -l i nearl y i ndependent,then wecannothave (n 1 s m 1 )= (n 2 s m 2 )= 0 for i ntegers n 1 ;n 2 ;m 1 ;m 2 w i th (n 1 ;m 1 ) 6 = (n 2 ;m 2 ), that i s, zeros and pol es of di erent -factors cannot cancel . T here i s no cancel l ati on am ong l ocal factors, si nce l ocalfactors can onl y have zeros and never pol es. N ow consi der cancel l ati on am ong zeros of l ocal factors and -factors. W e want to show that there are at m ost ni tel y m any cancel l ati ons. Suppose otherw i se. Si nce there are onl y ni tel y m any l ocalfactors and ni tel y m any -factors,an i n ni tude ofcancel l ati on woul d i m pl y that there are i n ni tel y m any cancel l ati ons am ong one l ocalfactor and one -factor. T he zeros of a l ocal factor are of the form i + 2k i log p , w here i i s the l ogari thm ofone ofthe roots ofW (p;X )= 0 chosen i n such a way that 0 = i < 2 i log p . Si nce an al gebrai c equati on has onl y ni tel y m any roots, an i n ni tude of cancel l ati onsi m pl i es that for som e com pl ex num ber and i n ni tel y m any i ntegers k we have (n( + 2k i log p ) m )= 0. C hoose 4 di erentsuch i ntegersk 1 ;:::;k 4 ,and l et 1 ;:::; 4 be the correspondi ng roots of . T hen we have

1 2 = 2(k1 k2 )n log p , 3 4 = 2(k3 k4 )n log p ,that i s,(k 3 k 4 )( 1 2 )= (k 1 k 2 )( 3 4
),w hi ch gi ves a l i near rel ati on am ong the zeros of ,contradi cti ng our assum pti on. H ence,i fthe i m agi nary parsofthe rootsof are Q -l i neari ndependent,the num berofzerosand pol es of D i n som e dom ai n coi nci des w i th the sum of the num bers of zeros and pol es ofal lfactors,up to som e bounded error,and our cl ai m fol l ow s.

E xam ples

In thi s secti on we gi ve exam pl es to show that our cl assi cati on i s non-tri vi ali n the sense that every case actual l y occurs.

E xam ple 1. T hesum P 1 n = 1 2 (n ) (n ) n s = (s) (s 1) (2s) (2s 2) correspondsto thepol ynom i al W (X ;Y )= (1+ Y )(1+ X Y ),w hi l e the sum P 1 n = 1 (n ) n s = (s) (s 1)corresponds to the rati onalfuncti on W (X ;Y )= 1 (1+ Y )(1+ X Y ) .
E xam ple 2. (a) Let (n) be the num ber of pri m e di vi sors of n counted w i th m ul ti pl i ci ty. T hen

P 1 n = 1 2 (n ) n s = Q p (1 + 2 p s (1 p s ))) correspondsto the rati onal functi on W (X ;Y )= 1 + 2Y
1 Y w i th m ai n part 1 + 2Y ,w hi ch i s not cycl otom i c. (b) Let G be the di rect product of three copi es of the H ei senberg-group,a / n (G ) the num ber ofnorm alsubgroups ofG ofi ndex n. T hen / G (s) =

P 1 n = 1 a / n (G ) n s
was com puted by Tayl or [START_REF] Taylor | Zeta FunctionsofA lgebrasand R esolution ofSingularities[END_REF]and can be w ri tten as a ni te product of -functi ons and an Eul er-productofthe form

Q p W (p;p s ),w here W consi stsof14 m onom i al sand W (X ;Y )= 1 2X 13 Y 8 ,w hi ch i s not cycl otom i c. E xam ple 3. (a)
Let G be the free ni l potent group ofcl ass two w i th three generators. T hen / G (s) can be w ri tten as a ni te product of -functi ons and the Eul er-product

Q p W (p;p s ),w here W (X ;Y )= 1 + X 3 Y 3 + X 4 Y 3 + X 6 Y 5 + X 7 Y 5 + X 10 Y 8 :
W e have W (X ;Y )= 1 + X 7 Y 5 ,w hi ch cl earl y does not di vi de W ,hence,w hi l e W i s cycl otom i c,W i s not. H ence,W i s not case 1 or 2. T heorem 1 i m pl i es that 7=5 i s an essenti alsi ngul ari ty of / G . D u Sautoy and W oodward [START_REF] Sautoy | Zeta functions of groups and rings[END_REF]showed that i n fact the l i ne < s = 7=5 i s the naturalboundary for / G . (b) N ow consi der the product

f(s)= Y p 1 + p s + p 1 2s
A gai n,the pol ynom i alW (X ;Y )= 1 + Y + X Y 2 i s not cycl otom i c,w hi l e W i s cycl otom i c.A gai n,T heorem 1 i m pl i esthat1=2 i san obstructi ng poi ntoff.H owever, the questi on w hether there exi sts another poi nt on the l i ne < s = 1=2 w hi ch i s an obstructi ng poi nt i s essenti al l y equi val ent to the R i em ann hypothesi s. W e have

f(s) = (s) (2s 1) (3s 1) (2s) (4s 2) R (s) Y m 1
((4m + 1)s 2m ) ((4m + 3)s 2m 1) ((8m + 2)s 4m ) ;

hence,i f hasonl y ni tel y m any zeroso the l i ne 1=2+ it,then the ri ghthand si de has onl y ni tel y m any zeros i n the dom ai n < s > 1=2,j = sj> ,hence,1=2 i s the uni que obstructi ng poi nton thi sl i ne.O n the otherhand,i f (s)hasi n ni tel y m any non-realzeros o the l i ne 1=2 + it,then every poi nt on thi s l i ne i s an obstructi ng poi nt for f (confer [ 1] ).

H ence,w hi l e forsom e pol ynom i al sthe naturalboundary can be determ i ned,we do not expect any generalprogress i n thi s case. T he pol ynom i al

W (X ;Y )= 1 + (X + X 2 + X 3 + X 4 )Y + X 5 Y 2
sati s esthe rel ati on W (X ;Y )= 1+ X 4 Y ,thati s, W i scycl otom i c,w hi l e W i snot. D u Sautoy an G runewal d [START_REF] Sautoy | Zeta functions of groups: zeros and friendly ghosts,A m er[END_REF]showed that i n the cycl otom i c expansi on ofW there are onl y ni tel y m any (n;m ) w i th c n ;m 6 = 0 and n + 1 m > 4,and that W (p;p s )= 0 has sol uti ons w i th < s > 4 for i n ni tel y m any pri m es,hence,W i s an exam pl e of type 4,and Z (s=3) has the naturalboundary < s = 4. (b) Let V be the cubi c vari ety x 1 x 2 x 3 = x 3 4 ,U be the open subset fx 2 V [ Z 4 : x 4 6 = 0g,H the usualhei ght functi on. D e l a B ret eche and Si r Sw ynnerton-D yer [START_REF]Sw ynnerton-D yer, Fonction zêta des hauteurs associ ee a une certaine surface cubique[END_REF] showed that Z (s)=

P

x2 U H (x) s can be w ri tten as the product of ni tel y m any -functi ons,a functi on hol om orphi c i n a hal f-pl ane stri ctl y l arger than < s > 3=4, and a functi on havi ng an Eul er-product correspondi ng to the rati onalfuncti on

W (X ;Y )= 1+ (1 X 3 Y )(X 6 Y 2 + X 5 Y 1 + X 4 + X 2 Y 2 + X Y 3 + Y 4 ) X 9 Y 3 :
T hey showed that i n the cycl otom i c expansi on of thi s functi on there occur onl y ni tel y m any term s c n ;m X n Y m w i th c n ;m 6 = 0 and n + 1 m > 3 4 ,and al lbut ni tel y m any l ocalfactors have a zero to the ri ght of< s = 3=4,hence,< s = 3=4 i s the naturalboundary ofZ (s).

E xam ple 5. Let J 2 (n) be the Jacobsthal -functi on, i . e. J 2 (n) = # f(x;y) :1 x;y n;(x;y;n)= 1g,and de neg(s)=

P n 1 (n )J2 (n ) n 2
.Si nceJ 2 i sm ul ti pl i cati ve, g has an Eul er-product,w hi ch can be com puted to gi ve

g(s)= Y p 1 + p s p 2 s : W e have g(s)= Y p (1 p 2 s ) Y p (1 + p s 1 p 2 s )= (s 2)D (s);
say.For = < s > 2 + the Eul er product for D convergesuni form l y,si nce

X p p s 1 p 2 s X p p 1 2 2
(2) :

H ence,D i s hol om orphi c and non-zero i n < s > 2,that i s,no poi nt on the l i ne < s = 2 i s an obstructi ng poi nt, that i s, Esterm ann' s m ethod cannot prove the exi stence ofa si ngl e si ngul ari ty ofthi s functi on.

C om parison of our classification w ith the classification of du Sautoy and W oodw ard

In [START_REF] Sautoy | Zeta functions of groups and rings[END_REF] ,du Sautoy and W oodward consi derseveralcl assesofpol ynom i al sforw hi ch they can prove C onjecture 1. Si nce thei r cl asses do not coi nci de w i th the cl asses descri bed i n T heorem 1,we now descri be how the two cl assi cati ons com pare.W e w i l lreferto the cl assesdescri bed i n T heorem 1 as' cases' ,w hi l e we w i l lconti nue to refer to the pol ynom i al s ofdu Sautoy and W oodward by thei r ori gi nalappel l ati on of'type' .

Pol ynom i al s oftype I are pol ynom i al s W such that W i s not cycl otom i c,thi s cl ass coi nci des w i th pol ynom i al s i n case [START_REF] Schlage-Puchta | M erom orphic C ontinuation of M ultivariable E uler P roducts[END_REF].

Pol ynom i al s oftype II are pol ynom i al s W such that W i s cycl otom i c,there are onl y ni tel y m any c n ;m > 0 w i th n + 1 m > ,and for i n ni tel y m any pri m es we have that W (p;p s ) has zeros to the ri ght of . T hi s cl ass contai ns al lpol ynom i al s i n case (4), and al lpol ynom i al s of type II fal l under case (3) or ( 4), but there are pol ynom i al s i n case (3) w hi ch are not oftype II. For pol ynom i al s oftype II they prove that the l i ne < s = i s the naturalboundary ofm erom orphi c conti nuati on ofD ,thei r resul t for pol ynom i al s therefore cl earl y supersedes the rel evantparts of T heorem 1.

Pol ynom i al softype IIIare pol ynom i al sW asi n type II,but there are i n ni tel y m any pai rs n;m w i th c n ;m > 0, n + 1 m > . T hese pol ynom i al s fal lunder case (3), they show under the R i em ann hypothesi s that < s = i s a naturalboundary. For such pol ynom i al s the resul ts are i ncom parabl e,our resul ts are uncondi ti onal ,yet weaker.

Pol ynom i al s oftype IV are pol ynom i al s w i th i n ni tel y m any pai rs (n;m ) sati sfyi ng c n ;m 6 = 0 and n + 1=2 m > ,and such thatw i th the excepti on of ni tel y m any p there are no l ocalzeros to the ri ght of< s = . For such pol ynom i al s du Sautoy and W oodward show that < s = i s the naturalboundary,i fthe i m agi nary parts ofthe zeros of are Q -l i nearl y i ndependent. A l lpol ynom i al s oftype IV fal lunder case (3),agai n,the resul ts are i ncom parabl e.

Pol ynom i al s oftype V are pol ynom i al s W such that W i s cycl otom i c,w i th the excepti on of ni tel y m any p there are no l ocalzerosto the ri ghtof ,and there are onl y ni tel y m any pai rsn;m w i th c n ;m 6 = 0 and n + 1 m . T hi s correspond to case [START_REF] Ahlquist | O n the analytic continuation of E ulerian products[END_REF].

Pol ynom i al softype V Iare pol ynom i al s W such that W i s cycl otom i c,w i th the excepti on of ni tel y m any p there are no l ocalzeros to the ri ght of , there are i n ni tel y m any pai rs (n;m ) w i th c n ;m 6 = 0 and n + 1 m > , onl y ni tel y m any of w hi ch sati sfy n + 1=2 m > . T hese fal lunder case (3). C ase (1) does not occur i n thei r cl assi cati on as i t i s justl y regarded as tri vi al .

C om parison w ith the m ultivariable case

T he objectofourstudy hasbeen the D i ri chl et-seri esD (s)= Q W (p;p s ). T hi s w i l lbe cal l ed the 1 1 2 -vari abl e probl em si nce the pol ynom i alhas two vari abl es,but the D i ri chl et-seri es depends on onl y one com pl ex vari abl e. If the coe ci ents of the above seri es have som e ari them eti cal m eani ng, and thi s m eani ng transl ates i nto a statem ent on each m onom i al of W , then the D i ri chl et-seri es D (s 1 ;s 2 ) = Q p W (p s1 ;p s2 ) retai ns m ore i nform ati on, and i t coul d be frui tful to consi der thi sfuncti on i nstead. O fcourse,the gai n i n i nform ati on coul d be atthe ri sk ofthe techni caldi cul ti es i ntroduced by consi deri ng severalvari abl es. H owever,here we show thatthe m ul ti vari abl e probl em i s actual l y easi erthen the ori gi nalquesti on of 1 1 2 -vari abl es. W here there i sno expl i ci treference to p,the probl em ofa naturalboundary was com pl etel y sol ved by Essouabri ,Li chti n and the rst nam ed author [START_REF] Schlage-Puchta | M erom orphic C ontinuation of M ultivariable E uler P roducts[END_REF] .

T heorem 2. Let W 2 Z[ X 1 ;:::;X k ] be a pol ynom ial satisfying W (0;:::;0) = 1. Set D (s 1 ;:::;s k ) = Q p W (p s1 ;:::;p s k ). T hen D can be m erom orphicall y continued to the whol e com pl ex pl ane if and onl y ifW is cycl otom ic. Ifit cannot be continued to the whol e com pl ex pl ane,then its m axim aldom ain ofm erom orphic continuation is the intersection of a nite num ber of e ectivel y com putabl e hal fspaces. T he bounding hyper-pl ane of each of these hal f-spaces passes through the origin.

A t rst si ght one m ay thi nk that one can pass from the 2-di m ensi onalby xi ng s 1 ,however,thi s destroys the structure ofthe probl em ,as i s dem onstrated by the fol l ow i ng.

E xam ple 6. T he D i ri chl et-seri es D (s 1 ;s 2 )= Q p 1 + (2 p s1 )p s2 as a functi on oftwo vari abl es can be m erom orphi cal l y conti nued i nto the set f(s 1 ;s 2 ) :< s 2 > 0;< s 1 + s 2 > 0g,and the boundary ofthi s set i s the naturalboundary ofm erom orphi c conti nuati on. Ifwe x s 1 w i th < s 1 0,and vi ew D as a functi on ofs 1 , then D can be conti nued to C i fand onl y i fs 1 = 0. In every other case the l i ne < s 2 = 0 i s the naturalboundary.

Proof. T he behavi our ofD (s 1 ;s 2 ) fol l ow s from [ 2,T heorem 2] . Ifwe x s 1 ,then 1 + (2 p s1 )p s2 has zeros w i th rel ati vel y l arge realpart,provi ded that ei ther < s 1 > 0,or < s 1 = 0 and < p s1 < 0. In the rst case we can argue as i n the case that W i snotcycl otom i c.B y the pri m e num bertheorem forshorti nterval swe nd thatthe num berofpri m enum bersp < x sati sfyi ng < p s1 < 0 i sgreaterthan c x log x , and we see that we can agai n adapt the prooffor the case W non-cycl otom i c.

In otherwords,the naturalboundary forthe 1 1 2 -vari abl e probl em i sthe sam e as for the 2-vari abl e probl em ,w i th one excepti on,i n w hi ch the 1 1 2 -vari abl e probl em col l apses to a 1-vari abl e probl em , and i n w hi ch case the Eul er-product becom es conti nuabl e beyond the 2-vari abl e boundary.

It seem s l i kel y that thi s behavi our shoul d be the preval ent one,i t i s l ess cl ear w hat preci sel y \thi s behavi our" i s. O ne qui te strong possi bi l i ty i s the fol l ow i ng: Suppose thatD (s 1 ;s 2 )= Q p W (p s1 ;p s2 ) has a naturalboundary at< s 1 = 0. T hen there are onl y nitel y m any val ues s 2 ,for which the special ization D ( ;s 2 ) is m erom orphicall y continuabl e beyond < s 1 = 0.

H owever,thi s statem ent i s ri ghtnow supported onl y by a generall ack ofexampl es,and thefactthatexam pl e6 l ooksqui tenatural ,so wedo notdarea conjecture. H owever we bel i eve that som e progress i n thi s di recti on coul d be easi er to obtai n than di rectl y handl i ng C onjecture 1. In parti cul ar those cases,i n w hi ch zeros of pose a seri ous threat for l ocalzeros woul d becom e a l ot easi er si nce thi s type of cancel l ati on can onl y a ect a countabl e num ber ofval ues for s 2 .

R eferences

[1] G . B how m ik, J.-C . Schlage-P uchta, N atural B oundaries of D irichlet series, Func. A pprox.

C om m ent. M ath.X X X V II.1 (2007),17{29.

m odul o 2 i

 2 i sa com pl ex num bers i n the segm entw hi ch i scongruentto z p1 i1 log p1 and congruentto w p2 i2 m odul o 2 i log p2 .G oi ng around thecycl eand col l ecti ng the di erences we obtai n an equati on ofthe form 2 i P i i log pi = 0, i 2 Z,w hi ch can onl y hol d i fthe com bi ned coe ci entsvani sh foreach occurri ng pri m e.H oweverthe coe ci ents cannot vani sh i fsom e pri m e occurs onl y once. Ifthe cycl e i s m i ni m al the sam e vertex cannot occur tw i ce, hence, there i s som e j such that p 1 = p j , but i 1 6 = i j . H ence, every m i ni m alcycl e contai ni ng z p1 i1 m ust contai n z pj ij or w pj ij

E xam ple 4 .

 4 (a) T he l ocalzeta functi on associ ated to the al gebrai c group G i s de ned asZ p (G;s)= Z G + p jdet(g)j s p d w here G + p = G (Q p )\ M n (Z p ), j : j p denotes the p-adi c val uati on and i s the norm al i sed H aar m easure on G(Z p ). In parti cul ar the zeta functi on associ ated to the group G = G Sp 6[START_REF] Igusa | U niversal p-adic zeta functions and their functional equations[END_REF]i s gi ven by Z (s=3)= (s) (s 3) (s 5) (s 6) Y p 1 + p 1 s + p 2 s + p 3 s + p 4 s + p 5 2s :

G autam iB how m i k,

Jan-C hri stoph Schl age-Puchta, U ni versi t e de Li l l e 1, A l bert-Ludw i gs-U ni versi t at, Laboratoi re PaulPai nl ev e, M athem ati sches Insti tut, U . M . R .C N R S 8524, Eckerstr.1, 59655 V i l l eneuve d' A scq C edex, 79104 Frei burg, France G erm any bhow m i k@ m ath. uni v-l i l l e1. fr jcp@ m ath. uni -frei burg. de