Cyril Branciard

Nicolas Brunner

Nicolas Gisin

Christian Kurtsiefer

Antia Lamas-Linares

Alexander Ling

Valerio Scarani

Testing quantum correlations versus single-particle properties within

come

Quantum theory predicts and experiments confirm that nature can produce correlations between distant events that are nonlocal in the sense of violating a Bell inequality [1]. Nevertheless, Bell's strong sentence Correlations cry out for explanations remains relevant. The maturing of quantum information science and the discovery of the power of nonlocal correlations, e.g. for cryptographic key distribution beyond the standard Quantum Key Distribution schemes [2,3,4], strengthen Bell's wish and make it even more timely.

In 2003, Leggett proposed an alternative model for nonlocal correlations [5], that he proved to be incompatible with quantum predictions. We present here a new approach to this model, along with new inequalities for testing it. Remarkably these inequalities can be derived in a very simple way, assuming only the non-negativity of probability distributions; they are also stronger than previously published Leggett-type inequalities [5,6,[START_REF] Paterek | Experimental Test of Nonlocal Realistic Theories Without the Rotational Symmetry Assumption[END_REF][START_REF] Branciard | Experimental Falsification of Leggett's Nonlocal Variable Model[END_REF]. The simplest of these inequalities is experimentally violated. Then we go beyond Leggett's model, and show that one cannot ascribe even partially defined individual properties to the components of a maximally entangled pair.

Formally, a correlation is a conditional probability distribution P (α, β| a, b), where α, β are the outcomes observed by two partners, Alice and Bob, when they perform measurements labeled by a and b, respectively. On the abstract level, a and b are merely inputs, freely and independently chosen by Alice and Bob. On a more physical level, Alice and Bob hold two subsystems of a quantum state; in the simple case of qubits, the inputs are naturally characterized by vectors on the Poincaré sphere, hence the notation a, b.

How should one understand nonlocal correlations, in particular those corresponding to entangled quantum states? A natural approach consists in decomposing P (α, β| a, b) into a statistical mixture of hopefully sim- * Electronic address: cyril.branciard@physics.unige.ch pler correlations:

P (α, β| a, b) = dλρ(λ) P λ (α, β| a, b) . (1)
Bell's locality assumption is P λ (α, β| a, b) = P λ (α| a)P λ (β| b); admittedly the simplest choice, but an inadequate one as it turns out: quantum correlations violate Bell's locality [1]. Setting out to explore other choices, it is natural to require first that the P λ 's fulfill the so-called no-signaling condition, i.e., that none of the correlations P λ results from a communication between Alice and Bob. This can be guaranteed by ensuring space-like separation between Alice and Bob. Non-signaling correlations happen without any timeordering: there is not a first event, let's say at Alice's side, that causes the second event via some spooky action at a distance. One may phrase it differently: nonsignaling correlations happen from outside space-time, in the sense that there is no story in space-time that tells us how they happen. This is the case in orthodox quantum physics, or in some illuminating toy models like the nonlocal box of Popescu and Rohrlich (PR-box) [START_REF] Popescu | Quantum nonlocality as an axiom[END_REF]. Mathematically, the no-signaling condition reads P λ (α| a, b) = P λ (α| a) and P λ (β| a, b) = P λ (β| b): the local statistics of Alice are not influenced by Bob's choice of measurement, and reciprocally.

In 2003, Leggett proposed another model of the form (1), which can also be experimentally tested against quantum predictions [5].

This model was recently brought into focus by the work of Gröblacher et al. [6]. The basic assumption of Leggett's model is that locally everything happens as if each single quantum system would always be in a pure state. We shall be concerned here with the case of binary outcomes α, β = ±1, though generalizations are possible. In this case, the supplementary variables λ in Leggett's model describe pure product states of two qubits, denoted by normalized vectors u, v on the Poincaré sphere:

λ = | u ⊗ | v , (2)
and the local expectation values have the usual form as predicted by quantum physics:

α λ = u| a • σ | u = u • a , (3)
β λ = v| b • σ | v = v • b . (4)
If the qubits are encoded in the polarization of photons, as in Leggett's initial idea, then the assumption is that each photon should locally behave as if it were perfectly polarized (in the directions u and v), Both in the original paper [5] and in [6], the model was presented by implicitly assuming a time-ordering of the events. Any model based on such an assumption had already been falsified by the so-called before-before experiment [START_REF] Stefanov | Quantum Correlations with Spacelike Separated Beam Splitters in Motion: Experimental Test of Multisimultaneity[END_REF][START_REF] Stefanov | Quan-tum entanglement with acousto-optic modulators: Twophoton beats and Bell experiments with moving beam splitters[END_REF], as Suarez emphatically stressed [START_REF] Suarez | Time and nonlocal realism: Consequences of the before-before experiment[END_REF]. However, assumptions (3)(4) clearly define non-signaling correlations, and Leggett's model can be defined without any reference to time-ordering. As a consequence, its study does add something to our understanding of quantum non-locality. But what exactly? In what are such P λ 's "simpler" than the usual quantum correlations? To answer these questions, we recall that, in quantum theory, the singlet state is such that the properties of the pair are sharply defined (the state is pure), but the properties of the individual qubits are not. In this perspective, Leggett's model is an attempt of keeping the correlations while reintroducing sharp properties at the individual level as well.

and
Leggett's model cannot reproduce the correlations of the singlet state. Experimental falsifications have already been reported, first under additional assumptions [6], then more directly [START_REF] Paterek | Experimental Test of Nonlocal Realistic Theories Without the Rotational Symmetry Assumption[END_REF][START_REF] Branciard | Experimental Falsification of Leggett's Nonlocal Variable Model[END_REF]. These works relied on the violation of so-called Leggett-type inequalities. Analog to Bell's inequalities, these criteria say that, under Leggett's assumptions (3) and (4), a measurable quantity L should satisfy L ≤ L max , while quantum theory predicts that L > L max can be observed for suitable measurements. An important feature of Leggett-type inequalities is that, contrary to Bell's inequalities, the bound L max is not a fixed number: instead, like the model itself, it depends on the quantum measurements that are performed. Consequently all experimental data aiming at disproving Leggett's model should present evidence that the settings used in the experiment have been properly adjusted.

All previously available derivations of Leggett-type inequalities were quite lengthy and failed to suggest possible improvements or generalizations. We have found a much more straightforward derivation (see the Methods section), simply based on the fact that each P λ must be a valid probability distribution, so in particular P λ (α, β| a, b) ≥ 0. Remarkably, this constraint of non-negativity of probabilities, weak as it may seem, is enough to induce an observable incompatibility between Leggett's model and quantum predictions. In our derivation, it also appears that the previously derived Leggetttype inequalities are sub-optimal; among the improved inequalities that our approach suggests, the simplest one 1: Alice's (green) and Bob's (red) measurement settings used to test inequality (5). In order for inequality (5) to hold, Bob's settings must be such that the three pairs ( bi, b ′ i ) form the same angle ϕ, and the three directions ei of bi -b ′ i (blue) are orthogonal. The best violation is obtained when Alice's settings ai are chosen to be along the directions of bi + b ′ i .

reads

1 3 3 i=1 |C( a i , b i ) + C( a i , b ′ i )| ≡ L 3 (ϕ) ≤ 2 - 2 3 | sin ϕ 2 | (5)
where C( a, b) = α,β αβP (α, β| a, b) is the usual correlation coefficient. This inequality holds provided the three measurements on Alice's side and six on Bob's fulfill some relations; a possible set of measurements is given in Figure 1. For the singlet state, quantum mechanics predicts

C Ψ -( a, b) = -a • b. Thus, for the settings just defined, L 3 (ϕ) is L Ψ -(ϕ) = 2| cos ϕ 2 | . (6)
This expression violates inequality (5) for a large range of values ϕ.

In order to test the Leggett-type inequality (5) in an experiment, we prepared polarization-entangled photon pairs into single mode optical fibers in a close approximation to a singlet state, similarly as in [START_REF] Branciard | Experimental Falsification of Leggett's Nonlocal Variable Model[END_REF]. In our setup (see Figure 2), we choose the settings of polarization measurements a i , b i and b i ′ for the individual photons by dialing in appropriate orientation angles γ A,B for two quarter wave plates (λ/4), and angles θ A,B for two absorptive polarization filters (PF). Details about the experimental implementation can be found in Supplementary Information I. Through four consecutive coincidence measurements between photodetectors D A,B for all combinations and coupled into single mode optical fibers (SMF), similarly as in [START_REF] Branciard | Experimental Falsification of Leggett's Nonlocal Variable Model[END_REF]. Polarization measurements in arbitrary settings for each photon are performed with polarization filters (PF) and quarter wave plates (λ/4), followed by single photon detectors (DA,B). More details can be found in Supplementary Information I.

of settings a, -a and b, -b, we establish an experimental value for a correlation coefficient C( a, b).

The correlation coefficients necessary to compose values for L 3 (ϕ = ±30 • ) were obtained with an integration time of T = 60 s per point, leading to values of 1.9068 ± 0.0009 for ϕ = -30 • and 1.9005 ± 0.0010 for ϕ = 30 • . This corresponds to a violation of the bound for L 3 (ϕ = -30 • ) and L 3 (ϕ = +30 • ) in Leggett's model by 83.7 and 74.5 standard deviations, respectively.

The asymmetry in the measured values of L 3 (ϕ) is an indication for experimental imperfections in the accuracy of the settings, such as a possible misalignment of one of the quarter wave plates with respect to the polarizing filters. To test this alignment, we collected values for L 3 over a larger range of ϕ with an integration time of T = 15 s per setting (Figure 3). The variation of L 3 with ϕ is compatible with the quantum mechanical prediction for a singlet state with residual colored noise and an orientation uncertainty of the quarter wave plate of 0.2 • .

The falsification of Leggett's model proves that it is impossible to reconstruct quantum correlations from hypothetical, more elementary correlations in which individual properties would be sharply defined. Let us argue that a much stronger statement holds, namely, that individual properties cannot even be partially defined.

We first consider the following straightforward generalization of Leggett's model: we allow the "local states" λ to be mixed states, e.g. photons with a degree of polarization η. So, we replace (3) and ( 4) by The derivation of inequalities for this model follows exactly the same pattern as for usual Leggett-type inequalities. In particular, the generalized version of ( 5) is

α λ = η u • a , β λ = η v • b , with 0 < η ≤ 1 . (7)
L 3 (ϕ) ≤ 2 - 2 3 η | sin ϕ 2 | (8)
which, for angles ϕ small enough, is violated by L Ψ -(ϕ) for any value of η > 0. Thus, as soon as the degree of purity of the "local states" is non-zero, this generalization of Leggett's model also fails to reproduce quantum mechanical predictions (see Supplementary Information II for a more complete analysis of this generalization). Experimentally, one cannot expect to conduct a meaningful comparison between those two predictions down to η = 0, due to imperfections in the state preparation. From the measurement of Figure 3, however, we can claim experimental evidence of a violation for all η ≥ 0.56, with a statistical significance of at least 3.65 standard deviations, thus putting a lower bound for this class of models (see Figure 4).

It is then natural to conjecture that no model of form (1), with non-signaling correlations P λ , can perfectly reproduce the correlations of the singlet state

C Ψ -( a, b) = -a • b, unless α λ = β λ = 0 (9)
for all measurements a and b (except perhaps for a few λ's in a zero-measure set). In the Methods section we prove this conjecture for models with discrete supplementary variables λ; continuity arguments should allow to extend the result to more general models. We thus have a necessary condition for a non-signaling model to be compatible with quantum mechanics. However, given a model with non-trivial marginals, finding an explicit inequality that can be tested experimentally against quantum predictions is another problem; for that, one needs the specific details of the model. In summary, with the general goal of improving our understanding of quantum correlations, we reformulated Leggett's model. No time-ordering of the events was assumed, and all assumptions were made on the local part of the correlations. We derived new Leggett-type inequalities, simpler and stronger than previously known. The simplest version of our inequalities has been experimentally violated. Finally we investigated more general models à la Leggett, for which we only imposed the nosignaling condition. We argued that any such model with biased marginals is incompatible with quantum predictions for the singlet state. This shows that quantum correlations cannot be reconstructed from "simpler" correlations in which the individual properties would be even partially defined. Nature is really such that, in some cases, individual properties are completely lost while global properties are sharply defined.

Our result is in good agreement with the recent work of Colbeck and Renner [START_REF] Colbeck | Hidden variable models for quantum theory cannot have any local part[END_REF], who have derived general inequalities to falsify such models with non trivial marginals. An example of a non-signaling model that successfully reproduces the singlet correlations can be found in [START_REF] Cerf | Simulating Maximal Quantum Entanglement without Communication[END_REF]; indeed this model has unbiased marginals. It is also worth mentioning the nonlocal model of Toner and Bacon [START_REF] Toner | Communication Cost of Simulating Bell Correlations[END_REF], which reproduces analytically the singlet correlations with one bit of communication. In this model, the probability distributions P λ have non vanishing marginals; however, the P λ 's are signaling. The remarkable property of the Toner-Bacon model is that the communication is cleverly hidden, such that the final probability distribution P is non-signaling. This work is part of the general research program that looks for nonlocal models compatible or incompatible with quantum predictions. The goal is to find out what is essential in quantum correlations. Here we found that in order to simulate or to decompose the singlet correlations, one can't use non-signaling correlations P λ with non-trivial marginals. This nicely complements the result of [START_REF] Brunner | Entanglement and non-locality are different resources[END_REF], that the correlations corresponding to very partially entangled states, hence large marginals, can't be simulated by a single PR-box, which has trivial marginals.

METHODS

A. Simple derivation of inequalities

that test Leggett's model

Convenient notations

As mentioned, in this paper we focus on the case of binary outcomes α, β = ±1. In this case, the correlations can conveniently be written as Throughout the Methods section, we shall use these notations; in the main text, we have used more standard and simplified notations, the correspondence being

P λ (α, β| a, b) = (10) 1 4 1 + α M A λ ( a, b) + β M B λ ( a, b) + αβ C λ ( a, b) .
α λ = M A λ ( a, b), β λ = M B λ ( a, b). The no-signaling condition is M A λ ( a, b) = M A λ ( a) and M B λ ( a, b) = M B λ ( b).
In order for the decomposition (1) to be a valid mixture of correlations, all distributions P λ should be nonnegative. As we said, this constraint is enough to derive Leggett-type inequalities. From eq. ( 10), one can see that the non-negativity implies the general constraints:

|M A λ ( a) ± M B λ ( b)| ≤ 1 ± C λ ( a, b) . (11)
Constraints on the marginals M A λ or M B λ thus imply constraints on the correlation coefficients C λ , and vice versa.

Let's now consider one measurement setting a for Alice and two measurement settings b, b ′ for Bob, and let's combine the previous inequalities (11) that we get for ( a, b) and ( a, b ′ ). Using the triangle inequality, one gets

|C λ ( a, b) ± C λ ( a, b ′ )| ≤ 2 -|M B λ ( b) ∓ M B λ ( b ′ )| . ( 12
)
These constraints must hold for all probability distributions P λ . After integration over the λ's, one gets, for the averaged correlation coefficients

C( a, b) = dλρ(λ)C λ ( a, b) |C( a, b) ± C( a, b ′ )| ≤ 2 -dλρ(λ) |M B λ ( b) ∓ M B λ ( b ′ )| . ( 13
)
This inequality is general for all models with "local marginals", i.e. that fulfil the no-signaling condition.

Derivation of a simple Leggett-type inequality

Now we derive an inequality satisfied by Leggett's specific model, that can be experimentally tested. Inequality [START_REF] Colbeck | Hidden variable models for quantum theory cannot have any local part[END_REF] implies, for the particular form of eq. ( 4) for Bob's marginals:

|C( a, b) + C( a, b ′ )| ≤ 2 -dλρ(λ) | v • ( b -b ′ )| . (14)
Let's consider three triplets of settings ( a i , b i , b ′ i ), with the same angle ϕ between all pairs ( b i , b ′ i ), and such that b i -b ′ i = 2 sin ϕ 2 e i , where { e 1 , e 2 , e 3 } form an orthogonal basis (see Figure 1). After combining the three corresponding inequalities [START_REF] Cerf | Simulating Maximal Quantum Entanglement without Communication[END_REF], using the fact that

3 i=1 | v • e i | ≥
1 and the normalization dλρ(λ) = 1, we finally get the Leggett-type inequality (5).

For a pure singlet state, inequality ( 5) is violated when |ϕ| < 4 arctan 1 3 ≃ 73.7 • , and the maximal violation is obtained for |ϕ| = 2 arctan 1 3 ≃ 36.9 • . In the case of imperfect interference visibility V ( LΨ -(ϕ) = 2V | cos ϕ 2 |), a violation can still be observed as long as V > V

(3) th = 1 -( 1 3 ) 2 = 2 √ 2
3 ≃ 94.3%. Note that other Leggett-type inequalities can be derived, as we mention in Supplementary Information III.

B. Any non-signaling model must have vanishing marginals

We prove here that all marginals in a non-signaling model must necessarily satisfy the constraints (17) and (18) below; and we argue that this in turn implies the claim made in the main text, namely: all the marginals must vanish (apart perhaps on a zero-measure subset of λ's).

In order for a general non-signaling model to reproduce all quantum correlations C Ψ -( a, b) = -a • b of the singlet state, one must have, according to equation ( 13), for all a, b, b ′ on the Poincaré sphere:

dλρ(λ) |M B λ ( b) ± M B λ ( b ′ )| ≤ 2 -| a • ( b ∓ b ′ )| (15)
and therefore, for all b, b ′ :

dλρ(λ) |M B λ ( b) ± M B λ ( b ′ )| ≤ 2 -|| b ∓ b ′ || (16)
where || • || is the euclidian norm.

In the case where b ′ = -b, the first constraint of eq. ( 16) implies that for all b

dλρ(λ) |M B λ ( b) + M B λ (-b)| = 0 . (17)
In the case b ′ → b, the two vectors being normalized, we

have 2 -|| b + b ′ || = 2 -4 -|| b -b ′ || 2 ∼ || b-b ′ || 2 4
and therefore the second constraint of ( 16) implies that for all b

lim b ′ → b dλρ(λ) |M B λ ( b) -M B λ ( b ′ )| || b -b ′ || = 0 . (18)
Let us now prove, in the case of discrete λ's (λ ∈ {λ i }), that (17) and (18) in turn necessarily imply that the marginals M B λ ( b) must vanish. In this case, the integral dλρ(λ) should be changed to a discrete sum i p λi . Constraint (17) implies indeed that for all λ i (such that p λi > 0) and for all b, M B λi (-b) = -M B λi ( b), i.e. M B λi must be an odd function of b. This is a very natural property that one would impose to such a model; in particular, Leggett's model has indeed odd marginals.

In addition, constraint (18) implies that for all λ i and for all b, lim b

′ → b |M B λ i ( b)-M B λ i ( b ′ )| || b-b ′ ||
= 0, i.e. that all M B λi are differentiable, and their derivative is 0 for all b: therefore they are constant. Since they have to be odd functions, then necessarily they are equal to zero.

In conclusion, for discrete λ's, Bob's marginals M B λi ( b) must all vanish; of course, the same reasoning holds for Alice's marginals M A λi ( a). This result should also be valid for any distribution ρ(λ), at least those physically reasonable (e.g. piecewise continuous): we conjecture that for any reasonable model to reproduce the quantum correlations of the singlet state, necessarily the marginals must vanish, in the sense that linear polarization bases both before and after the acquisition of the correlations for L 3 in the usual way (i.e., γ A = θ A fixed, γ B = θ B varies). Supplementary Figure 1 shows these polarization correlations directly before the main measurement. We extract visibilities of V HV = 98.9 ± 0.8% and V ±45 • = 97.8 ± 0.8%, respectively.

The uncertainties in all experimental values of L 3 are obtained assuming Poissonian counting statistics for photodetection events, independent variations for each setting, and the usual error propagation through (S3) and (5).

To track potential drifts of the apparatus, the set shown in Figure 3 of the article was recorded in two interleaving series with a spacing of 5 • for ϕ each. The absence of a modulation of L 3 (ϕ) with a 5 • period in the presented graph indicates that the residual asymmetry can not be attributed to drifts of our setup, but to a residual misalignment of the quarter wave plates with respect to each other. From a simulation with a noisy singlet state, the residual asymmetry seems to be compatible with a misalignment on the order of 0.2 • .

The polarization correlations after these measurements revealed values of V HV = 99.3 ± 0.8% and V ±45 • = 97.8±0.8%, indicating again that the alignment of source, optical fibers and measurement system has not drifted significantly over the measurement time of about 9 h.

  FIG. 1:Alice's (green) and Bob's (red) measurement settings used to test inequality(5). In order for inequality (5) to hold, Bob's settings must be such that the three pairs ( bi, b ′ i ) form the same angle ϕ, and the three directions ei of bi -b ′ i (blue) are orthogonal. The best violation is obtained when Alice's settings ai are chosen to be along the directions of bi + b ′ i .

FIG. 2 :

 2 FIG.2: Experimental setup. Polarization-entangled photon pairs are generated in a nonlinear optical crystal (BBO) and coupled into single mode optical fibers (SMF), similarly as in[START_REF] Branciard | Experimental Falsification of Leggett's Nonlocal Variable Model[END_REF]. Polarization measurements in arbitrary settings for each photon are performed with polarization filters (PF) and quarter wave plates (λ/4), followed by single photon detectors (DA,B). More details can be found in Supplementary Information I.

FIG. 3 :

 3 FIG. 3: Violation of Leggett's model. The experimental values for L3 over a range of separation angles ϕ (points with error bars) violate the bound given by Leggett's model (solid line), and follow qualitatively the quantum mechanical (QM) prediction (dashed line). The error bars indicate ±1 standard deviations of propagated Poissonian counting statistics assumed for photodetection events. The largest violations of inequality (5) are found for ϕ = ±25 • with 40.6 and 38.1 standard deviations, respectively.

FIG. 4 :

 4 FIG.4: Experimental test of the generalized Leggett inequality for different degrees of purity η. From the measurement shown in Figure3, we extract the maximal excess of the experimental L3(ϕ) with respect to the upper bound (denoted LL) in inequality[START_REF] Branciard | Experimental Falsification of Leggett's Nonlocal Variable Model[END_REF], for various degrees of purity η. For η < 0.6, data at ϕ = -15 • are considered, while for η > 0.6, the largest violation are observed for ϕ = -25 • .

  This expression allows one to clearly distinguish the marginals M A λ ( a, b) = α,β αP λ (α, β| a, b) on Alice's side and M B λ ( a, b) = α,β βP λ (α, β| a, b) on Bob's, and the correlation coefficient C λ ( a, b) = α,β αβP λ (α, β| a, b).

  for all a, b, |M A λ ( a)| = |M B λ ( b)| = 0 for "almost all" λ (except for a zero-measure subset of λ's, that could possibly depend on a, b).

upplementary Figure 1 :

 1 Polarization correlations in the HV basis (open circles, θA = 0 • ) and ±45 • linear basis (filled circles, θA = 45 • ). The corresponding visibilities are VHV = 98.9 ± 0.8% and V ±45 • = 97.8 ± 0.8%.

  the local observations, conditioned on each λ, should fulfill Malus' law. It is worth emphasizing that Leggett's assumption concerns exclusively the local marginals α λ and β λ of the probability distributions P λ , while nothing is specified about the correlation coefficients αβ λ .

	Leggett's
	model can thus still be nonlocal, and can in general vio-
	late a Bell inequality.

Acknowledgements

We thank A. Ekert and C. Simon for useful comments. C.B., N.B., and N.G. aknowledge financial support from the EU project QAP (IST-FET FP6-015848) and Swiss NCCR Quantum Photonics. A.L.L. and C.K. acknowledge ASTAR for financial support under SERC grant No. 052 101 0043. This work is partly supported by the National Research Foundation and Ministry of Education, Singapore.

Correspondence and requests for materials should be addressed to C.B.

SUPPLEMENTARY INFORMATION

Cyril Branciard, Nicolas Brunner, Nicolas Gisin Group of Applied Physics, University of Geneva, Geneva, Switzerland Christian Kurtsiefer, Antia Lamas-Linares, Alexander Ling, Valerio Scarani Centre for Quantum Technologies / Physics Dept., National University of Singapore, Singapore

SUPPLEMENTARY INFORMATION I: TECHNICAL ASPECTS OF THE EXPERIMENT A. Pair source and detection

The polarization-entangled photon pair source in our experimental setup (Figure 2 of the article) is based on a non-collinear type-II parametric down conversion process in a 2 mm thick properly cut Barium-beta-borate crystal, where photon pairs with almost degenerate wavelengths around 702 nm are collected [1]. As a pump source (PS), we used an Argon ion laser with a center wavelength of 351 nm and a power of 38 mW. Longitudinal and partial transverse walk-off compensation was established in the usual way using a half wave plate and two 1 mm thick BBO crystals (CC) of the same cut as the conversion crystal [2].

For the polarization measurements we used zero order quarter wave plates (λ/4) and polymer-film based polarization filters (PF) with a small wedge error and an extinction ratio of better than 1:10,000. All four elements could be oriented with motorized rotation stages with a reproducibility of about 0.1 • .

Photo events were generated with fiber-coupled passively quenched silicon avalanche photodiodes (D A,B ). All photo events were recorded, and pairs identified when photo events coincided in a time window of τ = 15 ns.

A pair of interference filters (IF, center wavelength 702 nm, full transmission width 5 nm at half maximum) was used for initial alignment of the pair source, and left in the system to avoid a change in beam orientations.

B. Alignment and setup characterization

As a first step in our alignment procedure, the single mode optical fibers were adjusted for neutral polarization transport with polarization controllers (FPC). Under stable temperature conditions, we maintained an extinction ratio better than 1:1000 over half a day. Then, the exact orientation θ B of the polarizing filter with respect to the first one was fixed to an accuracy of 0.2 • by searching for a minimum of coincidence events, with the quarter wave plates removed. Compensation crystals were adjusted to maximize the visibility of polarization (anti-)correlations in the ±45 • basis. At this point, we were reasonably confident that we observe photon pairs in a good approximation of a singlet polarization state. Typical visibilities we reached were V HV ≈ 99% and V ±45 ≈ 98%.

There is no direct evidence in the experimental result for L 3 (ϕ) if the orientation of the quarter wave plates with respect to the polarizing filters is accurate. We thus verified the orientation of the first wave plate by symmetrization of the polarization correlations with a fixed polarization filter in Bob's arm under 0 • linear polarization, and varying θ B while observing coincidences. With this procedure, we were able to find the symmetric position within ∆γ A ≈ 0.2 • . The quarter wave plate in Bob's arm was oriented in a similar way.

To test the inequality (5) of the article, we used the following Poincaré vectors (see Figure 1):

ϕ 2 e i , with ( e 1 , e 2 , e 3 ) = ( y, z, x). (S2)

The x axis in this notation corresponds to ±45 • linear, the y axis to circular, and the z axis to horizontal/vertical polarization, the latter coinciding with the natural basis of the parametric down conversion process in the nonlinear optical crystal.

The correlation coefficients C( a, b) for each of the six settings for testing inequality (5) were obtained by recording photo-detection coincidences c during an integration time T for four combinations of orthogonal polarizations by rotating the polarizers by additional 90 • accordingly:

This way, artefacts due to imbalanced detector efficiencies were minimized.

In order to ensure the stability of the alignment, we recorded polarization correlations in the HV and ±45 •

SUPPLEMENTARY INFORMATION II: OTHER LEGGETT-TYPE INEQUALITIES

Inequality (5) in the article is certainly the simplest one that allows to test Leggett's model versus quantum mechanics. It involves only 3 settings for Alice and 6 for Bob. Note that on Alice's side, two settings could actually be chosen to be the same: in our case, one could for instance rotate the triplet ( a 1 , b 1 , b ′ 1 ) around the y axis, so that a 1 = a 3 = z. Thus, 2×6 settings are actually enough [START_REF] Paterek | Experimental Test of Nonlocal Realistic Theories Without the Rotational Symmetry Assumption[END_REF]. In addition, inequality (5) is stronger than the previously derived inequalities [3,4,5,6]. Consider for instance the simplest inequalities of [5,6]: in our formalism, they would be obtained by (i) choosing four triplets, in two orthogonal planes, and (ii) setting a i = b i . Now, the choice of the triplets was not optimal, nor was the choice (ii), as the geometry of the problem clearly shows.

Following our reasoning, one can derive even more robust inequalities by increasing the number N of (judiciously chosen) triplets (

holds for all v, then one gets the Leggett-type inequality

This inequality is violated by the singlet state, for which

N . The next simple case after the one we studied is N = 4: in this case the four directions e i should point to the four vertices of the regular tetrahedron. One can then show that

, and one gets an inequality that tolerates a visibility of V (4) th = 5 6 ≃ 91.3%. In the infinite (and experimentally non-testable) limit of all directions e scanned by the vectors b i -b ′ i , we would have ξ N → 1 2 , and V

2 ≃ 86.6%.

SUPPLEMENTARY INFORMATION III: A STRAIGHTFORWARD GENERALIZATION OF LEGGETT'S MODEL

Here we come back to the generalization of Leggett's model, in which we impose that the marginals should have the following form:

This generalized model satisfies inequalities similar to (S4):

For any value of η > 0, these inequalities are violated by the quantum mechanical prediction L Ψ -(ϕ), for a small enough angle ϕ. Thus, such generalizations of Leggett's model also fail to reproduce quantum mechanical predictions, and can also be tested experimentally.

The required visibility to violate inequalities (S6) depends on η: V (N ) th = 1 -(η ξ N ) 2 . In the experimental run shown in Figure 3, we had N = 3, ξ 3 = 1 3 , and an average visibility of 98.4 ± 0.4% . Therefore our data allows us to falsify these generalized models for all


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From The Suppliants (Hiketides) to the Oresteia:

A unique relation between the music of Iannis Xenakis and the staging of Alexis Solomos

Andriana Soulele CRIHAM, University of Poitiers, France

Proceedings of the Xenakis International Symposium Southbank Centre, London, 1-3 April 2011 -www.gold.ac.uk/ccmc/xenakis-international-symposium Iannis Xenakis wrote the music for the representations of two Aeschylian tragedies: Hiketides (or The Suppliants), in 1964 for the Epidaurus Festival and Oresteia in 1966, showed in Ypsilanti, Michigan. They were staged by one of the most prolific personalities of the post-war, Alexis Solomos. Solomos' collaboration with Xenakis was one of the most creative ones, not only for the Hketides but for the Oresteia too. These two representations, which are similar in many points regarding the staging and the musical illustration, show a singular aesthetical approach and a remarkable fusion between the musical and the theatrical conception of the two artists. The study of Xenakis' correspondence with Solomos, the analysis of the scores and of the scores' sketches and the examination of unpublished texts and notes permit to disclose and understand the mutual exchanges and influences between the composer and the stage director, in matters of theatrical realization and of musical accompaniment, as well as their final choices. By revealing Solomos' attentions and Xenakis aims for the general aspect of the performances at first and then for particular subjectsespecially the philosophical dimension, the dramatic and musical interpretation of the lyrical parts -we can draw a singular picture of the creative procedure. Furthermore, the comparison between the Hiketides and the Oresteia can allow us distinguishing numerous theatrical and musical similarities, in a point where the first representation's major influence to the second one, can be certainly affirmed. In fact, Xenakis innovative ideas for the instrumentation, the concept of sound masses and the sound's spatial dimension initially introduced in the Hiketides were developed in a large scale in the Oresteia. On the other hand, Solomos's concept for renovating the Greek tragedy's staging by returning to the "primitive superstition" and to the "irrepressible shout of passion where music has an essential role" seems to be relevant for both performances.

In the 1960s, which were a very prolific period for the revival of ancient Greek tragedy in Greece, Iannis Xenakis wrote the music for the representations of two Aeschylian tragedies: Hiketides (or The Suppliants), in 1964 for the Epidaurus Festival and Oresteia in 1966, showed in Ypsilanti, Michigan. Staged by the eminent stage director Alexis Solomos, these productions revealed an innovating aesthetical concept both in staging and in musical composition. Guided by a prominent inspiration and the need for originality, Xenakis and Solomos paved the way for new fields of theatrical and musical experimentation. An undoubted reason for their triumphant work is certainly their creative artistic collaboration.

In the cases of theatrical productions, where music has inevitably a secondary role, the cooperation between the composer and the stage director, which is developed more or less in mutual interactions, choices and decisions, determines the final result, whether successful or not. Thus, writing music for the Greek tragedy's modern representation depends by and large on the general aesthetic conception of the stage management, which has to overcome several shortcomings. Therefore, the stage director's options characterized either by a preoccupation of archaeological exactitude, either by the use of modern scenic terms or by a desire of theatrical experimentation, influence directly the composer's work.

Alexis Solomos's point of view

The aesthetical convergence between Xenakis and Solomos was evident from the very beginning. Solomos's first letters addressed to Xenakis in February 1964 where he requested him to write the music of Hiketides, as their general correspondence for the representation, reveal the common points between the staging and the incidental music. Assisted by the sculptor Giannis Pappas, the costumer Giannis Moralis and the choreographer Agapi Evanguelidi, Solomos wanted to break the same old and conventional tradition of productions that were largely influenced by the aesthetical concepts of historians like Johan Joachim Winckelmann or stage directors like the Greek Dimitris Rondiris 1 . The return to the "primitive superstition" and to the "irrepressible shout of passion" where music has an essential role was his goal 2 . According to him, Xenakis's music was the only one that would give life to his innovating experiment [START_REF]σύγχρονη αντιµετώπιση του µουσικού προβλήµατος, που υψώνει µπροστά τραγωδία, πρέπει αραδιάζουν οι πηγές και να µη δογµατίζουµε µονολιθικά. φυσικά, υπόψει[END_REF] . Being aware of the musical problem that Greek tragedy's modern representation imposed, Solomos incited the use of imagination, the exception to the rules, and the absence of dogmas 4 . On his book What for Dionysus, he mentioned:

A brilliant musician knows that the sound of the ancient instruments isn't the same as with the sound of today's ones. Therefore, the choice and the combination of the usual strings, woodwind or brass instruments of a symphonic orchestra cannot be sufficient. On the contrary, he will invent and construct the right sound tools that will accompany the tragic chant, the declamation, the dance…When we were preparing Hiketides with Xenakis for Epidaurus and Oresteia for America, we agreed to the constitution of a sound orchestra with this character. 5 To Solomos, staging had to be submitted to music as music had to be submitted to staging. [START_REF]Letter of Alexis Solomos addressed to Xenakis[END_REF] His instructions and propositions regarding the character (e.g., melancholy, ritual, hysterical fear, choreographic pantomime), the action of every stasimon (choral part) and the appropriate music (e.g., collective songs, monotonous recitative like the parakatalogi [START_REF]A kind of melodramatic declamation accompanied by music[END_REF] of the byzantine tradition, accompanied declamation) for them, revealed his intention to perform the Hiketides choral parts as fanatic rituals of "primitive people" (exorcisms, invocations of the forces of the Moon and the Sun, etc.) and create prominent points where passion could be expressed by a collective hysteria, inarticulate screams and irrational vehemence [START_REF]Letter of Alexis Solomos addressed to Xenakis[END_REF] . He suggested the composition of chants capable of evoking the music of Polynesian (Maori), African or even Chinese people; the use of a chorus that evolves following the rhythm of a percussion instrument and even the creation of the Egyptian costumes armed with various objects and instruments like bells, triangles, woods or metals, in order to produce a strange and satanic sound when the Egyptians ran after the Danaids. 9

Iannis Xenakis's point of view

Solomos ideas, concepts and propositions exposed a particular chorus's treatment with massive movements and affirmed at the same time the primordial role of the music that seduced truly Xenakis. [START_REF]Letter of Iannis Xenakis adressed to Alexis Solomos[END_REF] The original theatrical score followed, enriched and incarnated Solomos's approach. Willing to see his music dominating for this representation, a wish that was partly shared by Solomos, Xenakis wrote the following to the stage director in a letter in April 1964:

Your wide and fresh mind allows me to hope that you will do anything in order to prevail the hierarchy of the music which, following the substance of your instructions, could sweep many parts of the tragedy to its current. [START_REF]καινούργιο και πλατύ σας πνεύµα µου επιτρέπει την ελπίδα ότι θα ασκήσετε πάν ό,τι ιεραρχία της µουσικής που ακολουθώντας το πνεύµα των οδηγιών σας[END_REF] Xenakis believed that simple means and new thoughts could actually provide a new plasticity to the Greek tragedy and revitalize its ritual character. [START_REF]Letter of Iannis Xenakis addressed to Alexis Solomos[END_REF] According to him, the Solomos's chorus conception was essential for Hiketides's representation because of its limited action: the combination of music and dance, the chorus playing percussion instruments that give the opportunity of dancing movements and the creation of total disorder in special moments could offer the essential and contemporary inhalation (breath) to the Greek tragedy. [START_REF]) η µουσική των χορωδειών διοχετεύονται απ'αυτόν. Γι'αυτό δύο αντίδοντα α. τα διάφορα κρουστά που συµµετέχουν στο χοροό και µπορεί να'ναι αφορµή χορευτικών κινήσεων[END_REF] Thus, keeping in mind the primordial role of the chorus, Xenakis wrote songs of symmetric rhythms and simple melodic lines of Greek traditional and archaic appearance, that had to be performed with plat voices, using a special vowels and consonants pronunciation; according to Solomos, it provided a real force to the words. 14 He reconsidered melodic (diatonic and chromatic tetracords, pedals points) and rhythmic structure of the Greek demotic and byzantine chant that survive from the antiquity [START_REF]Typed text titled « Notes sur l'Orestie[END_REF] for the vocal parts, while the instrumental music revealed his characteristic style with, among others, the use of micro intervals and numerous glissandi. The music for Hiketides intended to be produced by three different sources: a magnetic tape with the orchestra's part (2 trumpets, 2 trombones, 2 violins, 2 violoncellos, 2 contrabasses) diffused by loudspeakers around the theatre, live music by musicians located around the theatre's orchestra and a great number of percussions played by the female chorus 16 . Wooden, metallic and leather percussions instruments such as castanets, triangles, maracas, bells, tambours, etc., which were invented and designed especially for the representation of tragedy, were put on the costumes or played by the choristers during their dancing movements, their chant or their declamation, in order to displace the sources of sound production [START_REF]The chorus dances, sings or declaims being accompanied by various instruments in order to display the sound[END_REF] . For Xenakis, this kind of "auto-accompaniment" by the chorus provided new sound dimensions, new dancing movements and new massive movements as well.

According to him, the exploitation of these sound dimensions and the invention of percussion instruments could offer multiple directions to the tragic art in Greece [START_REF]Letter of Iannis Xenakis addressed to Alexis Solomos[END_REF] . That's particularly evident for this theatrical performance: by demanding to fix small instruments such as bells and sistrums onto the choristers' costumes (pockets, belts, etc.) and to put the bigger ones, such as tambours, around the set when they don't use them [START_REF]Letter of Iannis Xenakis addressed to Alexis Solomos[END_REF] , Xenakis creative and broad spirit influenced not only the sound production but the stage direction, the costumes and set design as well. Solomos approved widely Xenakis musical choices, the diversity of the percussion instruments and the composer's intention to create with the chorus movements "a mosaic of sounds: flexible, mobile and very rich, going from pianississimo to fortissisimo, from the synchronization and order to the total disorder"; 20 they were actually giving life to his proper staging instructions.

Relation between music and staging

In particular, Xenakis' notes for the realization of some choral parts recall the stage director's purposes very often. For instance, on the first stasimon of Hiketides (verse 524), the Danaids chorus evokes the powerful Jupiter, founder of their race that survived thanks to the descendants of Io. The composer put their words in music by using techniques such as the ison (pedal), diatonic tetracords and melodies in unison and in intervals of fourth and fifth, which evoke the structure of traditional Greek songs and byzantine psalmodies 21 . Solomos had suggested (in his letter in March 1964) for this choral part as for others, the use of a recitation accompanied by instruments, called parakatalogi that has survived through the byzantine time, in order to remind, by its monotony and absence of melody, the ascetic passion of a prayer. [START_REF]να χρησιµοποιήσω σε µερικά χορικά το recitative -αυτό που ονόµαζαν στα αρχαία χρόνια παρακαταλογή που επέζησε χάρη στο Βυζάντιο. Δεν πρέπει βέβαια να έχει βυζαντινό χαρακτήρα. µε τη µονοτονία του και την έλλειψη κάθε µελωδίας το ασκητικό πάθος µιας προσευχής[END_REF] Indeed Xenakis took this proposition into consideration by noting down at first on the same letter "katalogi in just one tone" [START_REF]καταλογή σε έναν τόνο µόνο χωρίς πτώσεις ή άρσεις της φωνής[END_REF] and by eventually composing an analogous chant for the chorus. He imagined the first part of this stasimon as a prayer where the chorus moves slowly, as a cortege, holding "door handles". Alluding to Easter's tradition where the Orthodox Christians hold candles as they walk, Xenakis suggested to Solomos a chorus that held "sound candles" as it displaced on stage, in a letter dated from mai 1964. [START_REF]Letter of Iannis Xenakis addressed to Alexis Solomos[END_REF] Considering also this choral part as a prayer to Jupiter that becomes progressively a kind of ritual as the choristers were referring to Io, Solomos stated to his instructions for Xenakis:

It's a kind of melancholic and emotional prayer, a collective, mystic song (chant) that could move Nietzsche to enthusiasm. (…) The stasimon's outline and appearance change while its rhythm accelerates without losing its religious character. The girls begin to represent Io's adventures as a part of a ritual. One of them is symbolically transformed to a cow while the others surround her. By a choreographic pantomime, they represent what they sing. [START_REF] Είναι | Οι κοπέλες αρχίζουν -σαν µέρος της ιεροτελεστίας -ν'αναπαρασταίνουν τις περιπέτειες της Ιώς. Μια απ'αυτές µεταµορφώνεται συµβολικά σε αγελάδα, κι οι άλλες την τριγυρίζουν. Ότι λέει το κείµενο που τραγουδούν, το αναπαραστούν µε παντοµίµα χορευτική[END_REF] In order to illustrate musically and ritually the second part of Jupiter's invocation, Xenakis conceived a singing interpretation, declamation and screams with non synchronous voices in total disorder with a rhythmical accompaniment by the orchestra and the chorus as well. He made clear to his notes that the beating of the kettledrums played by the choristers had to be disordered too, in a way to produce clouds of sounds. [START_REF] Aταξία | Η κάθε κόρη, φρενητικά κρούει η κάθε µια για τον εαυτό της σεβόµενη όµως τις σηµειωµένες εντάσεις[END_REF] As a result, he created chaotic sound masses that emphasized the mobility of sound and reinforced the primitive character of the prayer. On his sketches, he marked for this stasimon:

Collective song and dance, invocation to Jupiter: order ≤ disorder ≤ disorder, madness, screams, tambourines and metals and maracas, voices in different rhythms, cloudy declamation. [START_REF] Οµαδικό | επίκληση στο Δία : τάξη ≤ αταξία ≤ αταξία, τρέλλα, κραυγές, ντέφια κα µέταλλα κα µαράκες στριγλλά, φωνές σε διάφορους ρυθµούς, εκφώνηση ανακατωµένη[END_REF] In a letter addressed to Solomos in April 1964, he added that the chorus's hysteria and chaotic interpretation intended to produce a sound similar to the noise during the manifestations throughout the Italian and German occupation in Greece (between 1940 and 1945), after the confrontation with the enemy, where the slogans vanished into the chaos. [START_REF]σηµεία υστερίας όπου θα διασπαστούν, κι αφ'υπόδειγµα στην παρασήµανση αφ'φώνηση, τον εχθρό όπου τα συνθήµατα σπάνε σε χάος[END_REF] We can suppose that Xenakis was trying to reproduce musically his own experiences during the Second World War and the Greek Civil War that followed.

As a result, this choral part begun as a byzantine prayer when progressively the chant and the dance evolved giving their place to screams, panic movements and confused declamations. The continuum passage between order and disorder and the persistent pulsation of the drums as well created the ambiance of a primitive rite. His musical realization is in total harmony with the Solomos general perception of the Aeschylian choral parts -which he considered close to rituals characterized by invocations, magic, exorcisms, screams and hysterical behavior, [START_REF]Letter of Alexis Solomos addressed to Iannis Xenakis[END_REF] and especially with his particular concept of Jupiter's invocation: in his notes, Solomos emphasized the madness, the psychosis, the obsession and the frenzy of the Danaids chorus, too. [START_REF]καθαυτό ΟΜΑΔΙΚΟ ΧΟΡΙΚΟ (τραγούδι και χορευτική κίνηση µαζί) και βαστάει ίσαµε το 11. Νοµίζω πως οι έξι στροφές είναι πολλές -µπορούµε να τις περιορίσουµε σε 4. Πιθανότατα να χωρίσουµε τα ηµιχόρια. Το ένα να τραγουδά, ενώ το άλλο χορεύει και εναλλάξ. Αν και αυτό αφαιρεί την ένταση. Θα µ'σ'αυτό το χορικό τραγούδι -ή σ'αλλο παρακάτω-να προσθέταµε και να αφαιρούσαµε φωνές κλιµακωτά (από τη µια φωνή σε 42). Το τραγούδι αυτό και η όρχηση (10-11) θα προχωρούν µε δαιµονισµένη άνοδο του κοινού πυρετού. Ψύχωση, παραλήρηµα και τρέλλα θα συνεπαίρνει τις κοπέλες[END_REF] The obviously creative collaboration between Xenakis and Solomos for Hiketides's musical and theatrical performance put the foundations of their common work for the representation of the Oresteia in June 14 th 1966, in Ypsilanti, Michigan. The remarkable combination of compositional techniques that introduced the concept of the sound's spatialisation and the sound masse with a vocal writing illustrating the influence of traditional songs' and byzantine chants' melodies and rhythms, the invention of a large number of percussions, the use of traditional Greek instruments (such as the souravli) and the audacious staging transforming the tragic choral parts to vibrant, violent and passionate ceremonies that bring to mind Dionysian dithyrambs, [START_REF] Στασιμον Β | ΚΥΚΛΙΚΟΣ ΧΟΡΟΣ εξορκισµού και µαύρης µαγείας. Φαντάζοµαι πως πρόκειται για αναβίωση των παθιασµένων διονυσιακών διθυράµβων µιας παλιότερης εποχής[END_REF] were even more developed for this production of Aeschylus's trilogy. The Oresteia's concert version, Oresteia's Suite, that reproduces the majority of the original score with some modifications, reveals the primordial elements of this music, the esthetical concept of the composer as well as his theatrical thinking.

In particular, several choral parts disclose the affect of the byzantine chant, such as the verses 160, 176 of Agamemnon and the verse 152 of Libation Bearers. For instance for the verses 160 and 176, this is very obvious because of the use of syllabic style, the pedal (or isson) and the responsorial chanting between the chorus and the Coryphaeus: it reminds the religious chants between the first and the second priest of the Orthodox Church. According to the musical and theatrical draft notes for this choral part, both Xenakis and Solomos proposed an alternate interpretation in parakatalogi. However, the great interest of Oresteia's music resides also in the percussions instruments, in their significant role for the spatialisation of the sound and the musical illustration of the choral parts. Xenakis multiplied the number of the instruments used in Hiketides performance and their volume by delivering a large number of them to the orchestras' musicians, to the choristers and to the public, too. The chorus had to play bongos, tambours, metallic and wooden simandra (instrument used by the monks and the priests to the Greek monasteries and churches in order to call the Christians for the office), whips, sirens, metallic flags and maracas while the public had to use 200 small metallic flag papers. As a result, the music derived naturally from the dramatic interpretation, the actors' movements, the public's perception and the instrumental accompaniment. According to the composer himself, by giving the opportunity to play music to the choristers, music's mobility was directly linked to the staging 33 . This is mainly evident for some particular choral parts of The Eumenides, where the Erinyes described by Solomos as abstract nightmares and creations of Orestes's imagination had to act in a dynamic way according to the musical rhythm [START_REF]Τους βλέπω περισσότερο σανς δυσδιάκριτους ακαθόριστους εφιάλτες -δηµιουργήµατα της φαντασίας του Ορέστηπου στοιχειώνουν και τον βασανίζουν µε την αδιάκοπη παρουσία τους, πιο πολυ παρά µε τις επιθέσεις τους. ο ρυθµός το ζητά πρέπει να δρουν δυναµικά, γοργά κλπ. Αλλά µόνο και µόνο επειδή ο Ορέστης βρίσκεται σε έξαψη[END_REF] . The fact that Xenakis's musical imagination and Solomos's staging instinct were in total agreement encouraged for some of them an occult realization. For instance, Orestes's obstinate persecution by the Erinyes (verses 245-257) was illustrated by a talking text in recto tono accompanied by the coryphaeus' screams, sirens, chromatic and dissonant choral chant, brass winds' pedals and percussions' clouds. Both Solomos and Xenakis described this stasimon as a fast, nervous and savage choral part of an African and primitive character 35 . In addition, the Erinyes' invocation of the Night, their mother (verse 321), was perceived by Xenakis like a bewitchment [START_REF]Xenakis mentionned for the theatrical version of the verses 320-328 : « chanson collective, ensorcellement, incantation, danse[END_REF] , while Solomos imagined it as the revival of the passionate and Dionysian dithyrambs of the past, as an incantatory danse of black magic [START_REF] Κυκλικός | Φαντάζοµαι πως πρόκειται για αναβίωση των παθιασµένων διονυσιακών διθυράµβων µιας παλιότερης εποχής[END_REF] . The combination of the chant, the high-pitched screams, the sound clouds produced by whips, maracas and sirens and the orchestra's part, marked the wild side of the stasimon and demonstrated the theatrical concept of Solomos.

At last, the realization of the final choral part (verses 1033, 1044) celebrating the victory of justice reveals even more the remarkable convergence between composer and stage director: to Xenakis it was a collective chant, a cortege of musical explosions and to Solomos a triumphal deliverance, a lyric and optimist song interpreted by the crowd [START_REF]Στ. 1033-1047 -θριαµβευτική απολύτρωση και αισιοδοξία, εµφανίζονται πολλά πρόσωπα (+ ΑΝΔΡΕΣ) και γίνεται ένα λυρικό τραγούδι[END_REF] . In order to express the festive and chaotic ambiance, Xenakis associated the continuum, repetitive and gradually incomprehensible chant, the chorus's shouting, the orchestra's chromatic lines, glissandi and oscillations with the sound clouds of the percussions played simultaneously by the choristers, the musicians and the public. According to his notes, his goal was to create a real pandemonium and a crazy atmosphere with the attention of evoking a crowd in jubilation 39 .

Hiketides and Oresteia's performances reveal a unique relationship between the composition of the music and the staging direction, an authentic fusion of both Xenakis and Solomos perception of Greek Antiquity and musical-theatrical concept that led to exceptional innovations for Greek tragedy's modern representation : the mobility of the sound sources due to the displacement of the choristers playing several percussions on stage (which confers upon music a spatial dimension), the multiplication of the sound sources that results to the introduction of new concepts of stochastic music and the transformation of the dance to a kind of magic liturgy because of its enrichment with instruments and gestures [START_REF] Xenakis | Notes sur la Suite de l'Orestie[END_REF] . The unification of declamation, chant and danse and their intensification with non synchronous instrumental accompaniment and special vocal treatment (screams, shouting, etc.), that result often to noise, disorder and chaos, offered new means of massive expression to the tragic chorus [START_REF]ακριβώς και τέλεια συντονισµένα. επιτρέπει µια νέα πλαστική του ήχου (της µουσικής) τάξη στην αταξία και τανάπαλιν υπό τον στοχαστικό προσδιορισµό (determinisme ↔ indéterminisme), πράγµα που ανακάλυψα και εφάρµοσα υπό δεκαετίας στην ορχήστρα και στην ηλεκτρονική µουσική που έκανα[END_REF] .

Working on a rich field of experimentation and being inspired by extra-european cultures, Solomos showed how vivid, vibrant, powerful and original could be the staging of an ancient Greek tragedy without losing its context, nor its meaning. On the other hand, Xenakis' compositions demonstrated various influences and inspirations going from the ancient and traditional Greek music to avant-garde stochastic concepts that have made his work an inventive amalgam between the past and the future, the old and the new. Their artistic collaboration revealed a real aesthetical convergence that we can characterize, more or less, rare for such an imposing musical personality like Xenakis's and a visionary artist like Solomos; it couldn't be repeated in the performance of Oedipus at Colon in 1975, where the collaboration between Xenakis and Alexis Minotis failed because of their complete opposite interests and goals. Nevertheless, the prolific relation between Xenakis and Solomos gave birth to two pioneering representations that renewed in a unique way the contemporary production of the Greek tragedy, by emphasizing the implication between music and staging and by valorizing the genre of incidental music.

Sources

Iannis Xenakis Archives -Musical Department of the Bibliothèque Nationale de France

Figure 1 .

 1 Figure 1. Iannis Xenakis, Hiketides, verse 524, extracts from the first and third strophe (according to manuscript score, Archives Xenakis, National Library of France, BnF)

  32

Figure 2 .

 2 Figure 2. Iannis Xenakis, Oresteia, Agamemnon, verse 176, (Iannis Xenakis, Oresteia, for Baritone, Mixed and Children's Chorus, and Chamber Ensemble, Text from Aeschylus, Full Score, Boosey and Hawkes, 1996)

Gérard Condé, "Oresteia de Xenakis à Gibellina. Les temps bouleversés", Le Monde,Août 26, 1987. Dossier 5/4 : Écrits [Writings] Dossier 9/28 : Hikétides (Les Suppliantes) -écrits [Hiketides (The Suppliants) -Writings] Dossier 11/1-1(1) : Hikétides (Les Suppliantes) -partition manuscrite [Hiketides (The Suppliants)manuscrit score] Dossier 11/1-2: Hikétides (Les Suppliantes) -partition éditée [Hiketides (The Suppliants) -published score] Dossier 11/2 : Correspondance pour Hikétides (Les Suppliantes) -[Correspondence for Hiketides (The Suppliants)] Dossier 11/3 : Hikétides (Les Suppliantes) -schémas [Hiketides (The Suppliants) -diagrams] Dossier 11/4 : Photographies [Photographes] Dossier 11/5 : Coupures de presse [Newspaper excerpts] Dossier 13/1 : Orestie -Euménides [Oresteia -Eumenides] Dossier 13/2-1: Orestie -Agamemnon [Oresteia -Agamemnon] Dossier 13/2-2 : Orestie -Agamemnon[Oresteia -Agamemnon] Dossier 13/2-3 : Orestie -Agamemnon [Oresteia -Agamemnon]


Krzysztof Dominik Maślanka

Krzysztof Dominik

Extension of Maślanka Analytic

Keiper-Li 2022 Coe¢cients

Analytic Extension of

Keywords:

come   L'archive ouverte pluridisciplinaire

Introduction and de…nitions

One of the objectives of experimental mathematics is to look for hidden, unexpected patterns or structures using modern computational techniques. Of course, these experiments are not as valuable as the rigorous and immutable proofs that have been the goal and attribute of mathematics for centuries. But they can sometimes lead to the right track and suggest speci…c, useful directions for theoretical research -in the spirit of an ambitious program formulated as a kind of manifesto by Jonathan Borwein, an enthusiast and ardent propagator of experimental mathematics. In particular, Borwein recommends using computers to gain insight and intuition as well as discover new patterns and relationships ([4], page 2). And this is what we intend to do in this paper.

Here we will describe such an approach based on a speci…c example taken from analytic number theory. But …rst, let us recall some de…nitions and the notation adopted.

Function (s) introduced by Riemann in his fundamental paper [START_REF] Riemann | Ueber die Anzahl der Primzahlen unter eine gegebenen Grösse, Monatsberichte Königl[END_REF] (see also [START_REF] Harold | Riemann's Zeta Function[END_REF], p.16-21, for a detailed exposition):

(s) = 2(s 1)

s=2 1 + s 2 (s) (1)
is related to the zeta function (s), having the same complex (nontrivial) zeros n . All the so-called trivial zeros of the zeta function located at the negative even integers have been "annihilated" by the simple poles of the gamma function. (s) is an entire function 1 and satis…es simple functional equation:

(s) = (1 s) (2)
There are di¤erent conventions concerning the trivial factor in the de…nition of (s). Here we have chosen such that seems the most natural: (0) = (1) = 1.

Let us further de…ne:

f (s) = ln (s) (3)
This function is of crucial importance since it appears in the de…nition of certain real coe¢cients:

n = 1 (n) lim s!1 d n ds n s n 1 f (s) (4)
As was outlined by Keiper in 1992 [START_REF] Keiper | Power Series Expansion of Riemann's Function[END_REF] and rigorously proved by Li in 1997 [START_REF] Li | The Positivity of a Sequence of Numbers and the Riemann Hypothesis[END_REF] the celebrated Riemann hypothesis (RH) [START_REF] Riemann | Ueber die Anzahl der Primzahlen unter eine gegebenen Grösse, Monatsberichte Königl[END_REF] is equivalent to nonnegativity of coe¢cients (4):

RH () n 0 (5)
Of course, criterion [START_REF] Harold | Riemann's Zeta Function[END_REF], while deceptively simple, at least in its concise notation, is as di¢cult to settle as the Riemann hypothesis itself. Nevertheless it is worth detailed investigations, see [START_REF] Bombieri | Complements to Li's criterion for the Riemann hypothesis[END_REF], [START_REF] Johansson | Rigorous High-precision Computation of the Hurwitz Zeta Function and its Derivatives[END_REF], [START_REF] Je¤rey | Li coe¢cients for automorphic L-functions[END_REF], [START_REF] Maślanka | Li's Criterion for the Riemann Hypothesis -Numerical Approach[END_REF], [START_REF] Arias De Reyna | Asymptotics of Keiper-Li Coe¢cients[END_REF], [START_REF] Voros | From Asymptotic to Closed Forms for the Keiper/Li Approach to the Riemann Hypothesis[END_REF]. It should be emphasized that the coe¢cients n in the form (4), although important from theoretical point of view, are practically useless for numerical estimates.

Introducing function ' (s):

' (s) := 1 1 s (6)
one can easily show that n are power series coe¢cients of the logarithmic derivative of ' (s) (see [START_REF] Li | The Positivity of a Sequence of Numbers and the Riemann Hypothesis[END_REF] for details): '0 (s)

' (s) = 1 X n=0 n+1 s n jsj < 1 (7)
The third formula for n coe¢cients involves explicitly complex zeros of the Riemann zeta function located within the critical strip:

n = X 1 1 1 n ( 8
)
where the summation is taken over all these zeros with and 1 being paired together. A detailed inspection of the formula [START_REF] Je¤rey | Li coe¢cients for automorphic L-functions[END_REF] leads to a rather pessimistic conclusion: any attempt to locate a counterexample for RH by …nding a negative n is hopeless. Note that if = 1=2 + i y (for real y) then j1 1 j = 1. However, if the deviates from the critical line then j1 1 j is slightly di¤erent from one. Then n should be extremely great, especially for high , to make n negative.

Note that (8) can be thought of as an analytical extension of the coe¢cients n to an entire function (n), since n need not be an integer at all -it can be negative, rational, or even complex. The summation in [START_REF] Je¤rey | Li coe¢cients for automorphic L-functions[END_REF] would still converge over the entire complex plane. Unfortunately, this convergence is very slow -so slow that, for example, numerical calculation of the position of zeros of the entire function (n) is practically impossible.

Several important results on such an analytical extension of the Keiper-Li coe¢cients are contained in [START_REF] Je¤rey | Li coe¢cients for automorphic L-functions[END_REF], chapter 7. In this paper Lagarias rigorously proved its existence and also showed some of its properties.

There is yet another representation of these important constants, which we will give here just for the sake of completeness. Let

1 X j=0 a j s j (9)
where coe¢cients a j are all positive (see [START_REF] Li | The Positivity of a Sequence of Numbers and the Riemann Hypothesis[END_REF] for the proof). Then

n = n 1 X i=0 ( 1) i 1 i X a k1 :::a ki ( 10
)

Interpolation with nonequidistant nodes

Now our goal is to construct an interpolating polynomial convergent to f (s) ln (s) in the closed interval [0; 1]. To achieve this we choose as interpolation nodes the following set of points:

u = f1; 1 2 ; 1 3 ; 1 4 ; 1 5
; :::g

together with:

v = f0; 1 2 ; 2 3 ; 3 4 ; 4 5
; :::g

These nodes are obviously unequally spaced and get denser towards the edges of the interval [0; 1] on which we want to interpolate f (s). Note that with such choice, point 1 2 is doubled. Also note that each number from the set ( 11) is equal to one minus the corresponding number from the set [START_REF] Maślanka | Li's Criterion for the Riemann Hypothesis -Numerical Approach[END_REF], and vice versa which, of course, is intentionally related to the functional equation [START_REF] Philip Boas | Entire Functions[END_REF].

In the case of interpolation attempts, typical phenomenon is the so-called Runge e¤ect [START_REF] Trefethen | Approximation Theory and Approximation Practice[END_REF], which for equidistant nodes occurs even for some perfectly smooth (on the real axis) functions. These are undesirable oscillations between the nodes, growing with increasing order of the interpolating function. The remedy for this is the appropriate choice of unequally spaced nodes, e.g. so called Chebyshev nodes, which get denser towards the edges of the interval on which we want to interpolate a given function. It is this uneven distribution of nodal points that guarantees the convergence of the interpolation along the entire interval, and even in its neighborhood, but, of course, not the global convergence.

But …rst recall the de…nition and properties of the Pochhammer symbol (x) k which we shall need soon:

(x) k (k + x) (x) = k 1 Y i=0 (x + i) = ( 1) k k X i=0 ( 1) i S (i)
k x i = k X i=0 S (i) k x i (13)
where x is complex and k is a nonnegative integer. Let us now introduce the following family of polynomials of the variable s:

k (s) k Y i=1 s 1 i = s k k! s 1 s k = s k k! k X i=0 S (i) k s 1 s i (14) ! k (s) = k (s) k (1 s) = 1 (k!) 2 k X i;j=1 S (i) k S (j) k s k+i j (1 s) k+j i (15)
where

S (i)
k are (signed) Stirling numbers of the …rst kind (see [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] and [25]). The degree of the polynomial ! k (s) is 2k. As usual in such situations, we assume that if in [START_REF] Platt | The Riemann hypothesis is true up to 3 10 12[END_REF] any of the exponents is zero then the corresponding power is taken to be one. It is evident that ! k (s) is zero in k initial nodes (11) and [START_REF] Maślanka | Li's Criterion for the Riemann Hypothesis -Numerical Approach[END_REF]. Also, by construction, ! k (s) = ! k (1 s), i.e. for each k the functions ! k (s) have the same symmetry (2) as and f . Also, by construction the nodes (11), [START_REF] Maślanka | Li's Criterion for the Riemann Hypothesis -Numerical Approach[END_REF] are taken in pairs in [START_REF] Platt | The Riemann hypothesis is true up to 3 10 12[END_REF]. This is realized as follows: Such an interpolation mechanism is realized by taking a linear combination of polynomials [START_REF] Platt | The Riemann hypothesis is true up to 3 10 12[END_REF]:

k th order nodes 0 (0; 1) 1 (0; 1); (
F m (s) = m X k=1 ( 1) k k ! k (s) (16)
where m tends to in…nity and k are some coe¢cients that have yet to be determined. The condition from which we calculate them is F m (s) = f (s) if s takes the value of any of the nodes (11) or [START_REF] Maślanka | Li's Criterion for the Riemann Hypothesis -Numerical Approach[END_REF]. (Due to symmetry f (s) = f (1 s) nodes ( 11) and ( 12) are equivalent.) This condition leads to a system of linear equations: ::: ::: ::: ::: ::: :::

m F m (u m+1 ) or F m (v m+1 ) 1 1 4 1 = f 1 2 2 2 9 1 1 162 2 = f 2
1 C C C C C C C C C C C A = 0 B B B B B B B B B B B @
1 C C C C C C C C C C C A 0 B B B B B B B B B B B @ f 1 2 f 2 3 f 3 4 f 4 5 f 5 6
:::

1 C C C C C C C C C C C A (17)
Rational coe¢cients c kj which form the triangular matrix in [START_REF] Riemann | Ueber die Anzahl der Primzahlen unter eine gegebenen Grösse, Monatsberichte Königl[END_REF] were simply guessed which took some time and also required some intuition2 :

c kj = k j k!(k + 1) 2 j k (j + 1) 2k 2 1 j k k j 1 j j (18)
Finally we have:

k = k X j=1 c kj f j j + 1 (19)
As usual (cf. [START_REF] Maślanka | The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm[END_REF]) the most e¢cient way to obtain numerical values of k with high precision consists in tabulating at …rst appropriate values of f in the nodes with high precision (typically at least 10000 digits). In this way we avoid unnecessary, repetitive and time-consuming calculations of f in the nodes. A small but highly e¤ective program PARI/GP [START_REF][END_REF] is most appropriate to achieve this. Having these values one can use [START_REF] Trefethen | Approximation Theory and Approximation Practice[END_REF] to calculate k . It turns out that they decrease very quickly to zero as k grows, and their signs alternate (at least for the values available for numerical experiments, cf. Fig. 1.) Figure 1: Real coe¢cients k (19) diminish quickly to zero when k grows which guarantees fast convergence of series [START_REF] Arias De Reyna | Asymptotics of Keiper-Li Coe¢cients[END_REF]. Numerical calculations suggest that ( 1) k+1 k are all positive, at least if the Riemann hypothesis is true.

Numerically e¤ective formula for n

We are now ready to …nd numerically e¢cient expression for the Keiper-Li coe¢cients (4). Substituting the interpolating polynomial [START_REF] Arias De Reyna | Asymptotics of Keiper-Li Coe¢cients[END_REF] together with [START_REF] Platt | The Riemann hypothesis is true up to 3 10 12[END_REF] in place of f (s) in ( 4) and letting m ! 1 we get:

n = 1 (n) lim s!1 d n ds n 2 4 s n 1 1 X k=1 ( 1) k k (k!) 2 k X i;j=0 S (i) k S (j)
k s k+i j (1 s)

k+j i 3 5 = (20) = 1 (n) 1 X k=1 ( 1) k k (k!) 2 k X i;j=0 S (i) k S (j) k lim s!1 d n ds n h s n 1+k+i j (1 s) k+j i i
Using Leibniz' formula:

d n ds n (f (s)g(s)) = n X r=0 n r d r ds r f (s) d n r ds n r g(s) (21)
we get:

n = 1 (n) 1 X k=1 ( 1) k k (k!) 2 k X i;j=0 S (i) k S (j) k lim s!1 n X r=0 n r d r ds r (1 s) k+j i d n r ds n r s n 1+k+i j
The n th derivative of monomial is:

d n ds n (s c) r = r! (r n)! (s c) r n n r (22)
so

n = 1 (n) 1 X k=1 k (k!) 2 k X i;j=0 S (i) k S (j) k n X r=0 n r ( 1)
j i (k + j i)! (k + j i r)! (n 1 + k + i j)! (k + i j + r 1)! lim s!1 (s 1) k+j i r s k+i j+r 1 (23)
Now, after taking the limit s ! 1, in the summation over r survives only the term for which the exponent in (s 1) k+j i r is zero, i.e. when r = k + j i. Hence:

n = n 1 X k=1 k (k!) 2 k X i;j=0 S (i) k S (j) k ( 1)
j i 1 (n k j + i)! (n 1 + k + i j)! (2k 1)! = (24) = n 1 X k=1 k (k!) 2 k X i;j=0 S (i) k S (j) k n + k + i j 1 2k 1
De…ning rational coe¢cients nk :

nk = n (k!) 2 k X i;j=1 S (i) k S (j) k n + k + i j 1 2k 1 (25)
we …nally have:

n = 1 X k=1 nk k (26)
The rational coe¢cients nk are positive for integer n and increase with the increase of k. Nevertheless, because the coe¢cients k tend to zero much faster (see Fig. 1), the series representation (26) of n converges quite quickly. Therefore, it can be e¤ectively used to calculate several thousand of initial values of n , provided that we have previously tabulated numerical values of k , according to the procedure outlined above. For instance, if we have precomputed values of f (s) in the nodes with precision 20; 000 digits we can compute k up to k = 3361, since, as usual, precision of k decreases to zero with k. In order to increase this range, one should also (proportionally) increase the precision of the precomputed values of f (s) in the nodes.

The numerical values of coe¢cients n calculated in this way agree perfectly with values obtained by other methods, see e.g. [START_REF] Keiper | Power Series Expansion of Riemann's Function[END_REF], [START_REF] Johansson | Rigorous High-precision Computation of the Hurwitz Zeta Function and its Derivatives[END_REF], [START_REF] Maślanka | Li's Criterion for the Riemann Hypothesis -Numerical Approach[END_REF].

One thing should be stressed out here. In the formula (26), it is enough to sum up to some maximum k, more precisely, to that k for which the precision of k drops to zero. Most importantly, all digits of n obtained from this summation are signi…cant ( [START_REF] Maślanka | The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm[END_REF], for details see comment on formula [START_REF][END_REF] and especially Fig. 9).

Analytic extension of n

In what follows we are going to extend the range of index n from positive integers to arbitrary complex numbers. Simply note that the values of the parameter n in (26) no longer need to be positive integers as in the formula (4); on the contrary: they can be negative, rational and even complex. Thus (26) is in fact an extension of the formula (4) to the whole plane of complex numbers. For example:

(1=2) = 0:005774507219796948948:::

(i) = 0:02310189639985490400::: (1 + i) = 0:00001237486681209165::: + i 0:04619760137033736709:::

Our purpose is to demonstrate that (n) expressed by ( 26) is an entire function, so it has complex zeros that can be investigated numerically. It is also, as we will show, an even function, which is not obvious when looking at (4) or (26).

As mentioned above, the variable n is no longer limited to integers. This is because in the formula (25) it appears in a binomial symbol and not, for example, as an index in the Stirling numbers, where it should be integer by its very nature. Therefore we can use the following identity:

x k = ( 1) k k! ( x) k = 1 k! k X p=0 S (p) k x p (27)
and put x = n + k + i j 1: After some tedious but elementary calculations we get:

nk = n (k!) 2 1 (2k) 2k 1
X q=1;3;5;:::

kq n q (28) where kq = k X i;j=1 2k 1 X p=q S (i) k S (j) k S (p) 2k 1 p q (k + i j 1) p q (29)
(As usual, the power in (29) should be taken 1 when p = q.) It is easy to check that all the coe¢cients kq are integer and form a triangular matrix. Moreover, we see from (28) that nk are actually polynomials in variable n with rational coe¢cients. Below there are several of them labelled by the integer parameter k: In a naturally way, the integer parameter k enumerates consecutive polynomials nk , whereas n is an independent (in general complex) variable. Each nk is of degree 2k and has, of course, 2k roots. The distribution of these complex roots, even if not particularly interesting from the theoretical point of view, is quite aesthetic, see Fig. 2.

k nk 1 n 2 2 1 12 n 2 1 + 2n
Since kq = 0 for even values of q we have only even exponents of n q in (28). Introducing real coe¢cients q :

q = 1 X k=1 k;q 1 (2k) k (k!) 2 (30)
we …nally have (n) in the form of a "polynomial of in…nite order", i.e. the entire function (n) that interpolates n at integer values:

(n) = 1 X q=1 2q n 2q (31)
One can easily prove that coe¢cients q (30) with even q = 2; 4; 6; ::: have alternating signs and tend quickly to zero whereas those with odd q = 1; 3; 5; ::: are all exactly zero: In the summary of the considerations made so far, it can be stated that formula (31) works well for complex variable n. It is also clear that n = (n) is an entire even function, where n, as mentioned above, no longer needs to be limited to integers. Therefore, from now on, we will make change in the notation: n ! s.

We are quite aware that with extensions of this type there always exists a subtle problem of unambiguity: is the proposed extension unique? For instance, it is well-known that in the relatively simple case of extension of the ordinary factorial to an analytic function there are in…nitely many possibilities. However, the integral proposed by Euler is the simplest, most natural.

So quite consciously, we will remain faithful to numerical experiments that e¤ectively support intuition, in the spirit of the program presented by Jonathan Borwein. And, for the time being, we will leave the question of mathematical rigor.

Distribution of zeros

According to the general theory of entire functions [START_REF] Yakovlevich | Distribution of Zeros of Entire Functions[END_REF] expression (31) has complex zeros which may be investigated numerically. To achieve this the function FindRoot implemented in Wolfram's Mathematica is particularly useful as well as rich graphical capabilities of this remarkable program. For example, choosing the appropriate starting point s 0 in FindRoot[ (s),{s; s 0 },WorkingPrecision ! 100] one can …nd several thousand complex zeros of (s) (see Appendix below).

The …rst glance at the distribution of complex zeros of (s) reveals four symmetrical branches plus one double zero in the center (Fig. 3). However, logarithmic rescaling of the real axis makes the four branches almost straight lines (Fig. 4). The mechanism of the formation of a few initial zeros is shown in Figs. 5 and6. Numerical experiments clearly suggests that the growth of Re k with k is nearly linear while growth of Im k is almost perfectly logarithmic. Combining these dependencies we obtain simple formula:

Im k 16 log(Re k ) (32)
(Factor 16 is from the …tting procedure.) More detailed inspection of these zeros reveals certain surprising structure in their distribution (Figs. 7 and8). Another unexpected e¤ect which can be grasped only by computer experiments is certain strong, long-range correlation between values of the real and imaginary part of the zeros. This can be demonstrated by simply taking high-order …nite di¤erences, e.g. of the order of 700 or more. We'll give just two examples here. These are described in details below, in Figs. 9 and 10. Many other examples of such correlations can also be given, which is currently being investigated and will be the subject of another publication. Using Jensen's theorem, an independent test of the numerical correctness of several dozen initial zeros was carried out. By n(r) we denote the number of zeros of a given integer function:

f (s) = 1 X n=0 c n s n (33)
in a disk of radius r, jsj r, and by N (r) the integral:

N (r) = r Z 0 n(t) t dt with n(0) = 0 (34)
Jensen's theorem (cf. [START_REF] Philip Boas | Entire Functions[END_REF], §1.2.1) relates the distribution of zeros of an entire function to its growth. It states that: Both surfaces intersect the plane (x; y) (green color) along certain complicated curves. These curves in turn intersect themselves in certain de…nite points marked by vertical dark red lines, that is, in the complex zeros of (s). By elementary calculations we get that for (33) the following relation holds:

N (r) = 1 2 2 Z 0 ln f re i' d' ln jf (0)j (35)
n(r) = r 2 2 Z 0 h(r; ')d' (36)
where h(r; ') 1 2

d dr ln f (r; ') 2 + g(r; ') 2 (37)
and

f (r; ') 1 X k=1 c k r 2k 2 cos [2(k 1)'] (38)
g(r; ')

1 X k=1 c k r 2k 2 sin [2(k 1)'] (39)
Figure 11 illustrates how the formula (36) works. (Of course, in (38) and (39) the coe¢cients c k have been replaced by (30). In order to satisfy condition n(0) = 0 the series (31) has been divided by n 2 .)

(s) as a product over its zeros

Note that the complex zeros k of the lambda function introduced in this work are in a sense "dual" to the complex zeros n of the Riemann zeta function (or the xi function ( 1)) in the sense that both these families of zeros contain the same and complete information about this extremely important function.

Since (s) is an entire function, we can represent it, by Weierstrass's theorem, as an in…nite product over its zeros:

(s) = const s 2 Y all zeros 1 s (40)
(No additional prime factors are needed here because (s) is of …nite order [START_REF] Je¤rey | Li coe¢cients for automorphic L-functions[END_REF].) Remembering that lambda zeros come in fours (complex conjugate, di¤erent signs, i.e. k ; k ; k ; k ) we have:

(s) = const s 2 1 Y k=1 1 s k 1 + s k 1 s k 1 + s k = (41) = const s 2 1 Y k=1 0 @ 1 + s 4 2s 2 (Re k ) 2 (Im k ) 2 j k j 4 1 A
The constant in (41) can be calculated with high accuracy. Note that it is fairly close to (but of course di¤erent from) the …rst Keiper-Li coe¢cient: const = 0:023098802283424104776762431610::: This is because the …rst zero is relatively large: j 1 j = 104:7027678::: Formula (41) is based on the assumption that zeros k occur in fours. At least this is the case for small values of these zeros. Of course, we do not know if such distribution of zeros will persist on the global scale. In any case, this con…guration of zeros guarantees that the Riemann hypothesis will be true. Its violation, as we shall see, is associated with a radical deviation from the logarithmic dependence (32). Then, instead of complex fours of zeros, there will only be pairs of real zeros di¤ering in sign.

Violation of the Riemann hypothesis?

Finally, by analyzing the formula (41) from the point of view of the arrangement of complex zeros k , we can consider a hypothetical violation of the RH. As mentioned before, all attempts to use the Li's criterion [START_REF] Harold | Riemann's Zeta Function[END_REF] for this purpose are rather hopeless: in order to determine that the k th zero of Riemann zeta deviates from the critical line, one would have to …nd a negative coe¢cient n . As estimated by Oesterle [START_REF] Oesterlé | Régions sans zéros de la fonction zêta de Riemann[END_REF], such n would be of the order k 2 . Since it is known that there is no counterexample to the RH up to k 1; 236 10 13 (i.e. height of the zero 3 10 12 ) [START_REF] Platt | The Riemann hypothesis is true up to 3 10 12[END_REF], and most probably much higher, that means that all n up to n 1; 528 10 26 are nonnegative. It is then clear that direct searching for the breaking of the RH using Keiper-Li criterion is a hopeless task 3 . Such a hypothetical negative coe¢cient n would clearly mean that some factor in the product (41) should also be negative. It is easy to check by elementary calculations that this is impossible as long as we have four symmetrical zeros: k ; k ; k ; k (cf. Figures 3 and4). But if the logarithmic relationship Im k 16 log(Re k ) were broken, then -as the numerical experiments clearly suggest -zeros k would come closer to the real axis and ultimately there would be only two kinds of them: + k and k . This behavior would cause the coe¢cient n to become negative for some large index n due to oscillations increasing with n. This phenomenon is illustrated in Figures 12 and13.

Anyway, the resolution of Riemann's hypothesis is still as far away as it was when it was formulated over a century and a half ago.

Figure 12: The distribution of zeros k in the case of the truth of the Riemann hypothesis is marked with red dots. (Only one quarter of the complex plane is visible here.) Suppose, however, that somewhere far away there is a complex zero of the Riemann zeta function n that deviates from the critical line. (Of course, there would then be 4 such zeros: n , 1 n , n , 1 n .) The distribution of zeros of the function (s) in this (extremely) hypothetical situation is illustrated by blue points. For the initial values of n the distribution of blue points does not di¤er much from the distribution of red points. But as n increases, there will appear oscillations, followed by a sudden drop of the zeros k to the real axis. Eventually, starting with some k, all zeros will line up exactly on the real axis. Then there will be only two types of them: + k and k .

Figure 13: The …gure shows the reaction of coe¢cients n to the behavior of zeros k caused by the deviation of even single complex zero k of the Riemann zeta function from the critical line. The red dots here illustrate the distribution of n in case RH is true. (This distribution is described by an approximate law, see [START_REF] Maślanka | Li's Criterion for the Riemann Hypothesis -Numerical Approach[END_REF], eqn. 6). However, as n increases, rising oscillations appear and eventually n becomes negative. This will happen for n equal exactly to that k, from which the zeros k are already purely real (see previous Figure 11). 9 Appendix: Tables of complex zeros of (s)

Tables attached at the end of this article contain numerical values of the …rst 3520 zeros k with precision of 14 signi…cant digits. Note that each number x + iy in this table represents in fact four complex zeros x iy located symmetrically with respect to both axes of the complex plane. There is also one double zero 0 = 0.

k k k k k k k k 1 76.010927161420

Figure 2 :

 2 Figure 2: Distribution of complex roots of polynomials nk , labelled by the discrete parameter k, with n as a variable, in the range of k = 1; :::; 100. A given color represents a speci…c value of k, e.g. the outermost red dots correspond to k = 100.

Figure 3 :

 3 Figure 3: Distribution of 3665 zeros k of (s) on the complex plane. The scale on both axes is linear but with di¤erent units.

Figure 4 :

 4 Figure 4: The same points as in the previous Figure 3, but now the horizontal axis has a logarithmic scale.

Figure 5 :

 5 Figure 5: Contour plots of the real (upper) and imaginary (lower) part of (x + iy). Complex zeros k of (s) are marked by red dots.

Figure 6 :

 6 Figure 6: Three-dimensional view of the real (orange) and imaginary (blue) parts of (s) = (x + iy).Both surfaces intersect the plane (x; y) (green color) along certain complicated curves. These curves in turn intersect themselves in certain de…nite points marked by vertical dark red lines, that is, in the complex zeros of (s).

Figure 7 :

 7 Figure 7: Distribution of 3665 complex zeros k of (s). Vertical axis is rescaled as exp(Im( )/16). Amplitude of oscillations is multiplied by a factor of 5 for their better visualization.

Figure 8 :

 8 Figure 8: Unexpected distribution of complex zeros n in the range n = 1500; :::; 3665. It resembles something like "three-phase electric current" or -loosely speaking -a triple helix. In order to visually enhance the e¤ect, the colors of the points were selected accordingly: zeros number 1500; 1503; 15006; ::: are red, 1501; 1504; 15007; ::: are green and 1502; 1505; 15008; ::: are yellow.

Figure 9 :

 9 Figure 9: Another example of long-range correlations among zeros k of the complex function (s).Finite di¤erences of the order of 700 for the sequence of 3665 zeros k (left). On the right side we have the same, but a random perturbation -a complex number with an absolute value of no more than 4 10 5 -has been added to each zero. The destruction of the regular pattern is clearly visible.

Figure 10 :

 10 Figure 10: Finite di¤erences of the order of 700 for the sequence of 3665 zeros k of the entire function (s) but now every second zero was taken (left) or every fourth zero was taken (right).

Figure 11 :

 11 Figure 11: Whenever r exceeds the absolute value of the successive zero j k j, the value of n(r) given by (36) jumps by 4 as expected.

Strictly speaking, Riemann considered another function which is related to (1), namely ( 1

+ it). Since it was sometimes confusing, Edmund Landau introduced the present convention.

Explaining in detail how the formula(18) was obtained would be uninformative and, worse, very boring. There was a lot of guesswork, and there was also trial and error. Not very ambitious, but still e¤ective. First, the formula for the numbers from the main diagonal of (17) was guessed. Frequent consultation with the On-Line Encyclopedia of Integer Sequences[START_REF] Sloan | The On-Line Encyclopedia of Integer Sequences[END_REF] was very helpful at this stage.

It should be stressed out that the Oesterle's estimation is only asymptotical. For example, if we consider quite absurd situation that already the …rst zero of zeta is o¤ the critical line, then -depending on how much it deviates from this line -the …rst n that should "feel" this deviation and become negative would be for n equal approx. 7500.