Ali Assoudi

Amani Amamou

Nejla Mahjoub Saïd

Hervé Bournot

Characteristics and analysis of a turbulent offset jet including the effect of density and offset height

Keywords: Offset jet, Laser Doppler Velocimetry, Numerical simulation, Offset height, Variable density, Reynolds stresses ρ Local density Kg/m ρ j Flow ejection density Kg/m ρ ∞ Ambient density Kg/m

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Turbulent offset jet flows are obtained when a jet is discharged into a medium above a wall parallel to the axis of the jet exit but offset by a certain distance. The jet flow may be discharged with a variable density which may be due to temperature variations or variations in the composition by fluids of different density. Given their unique and complex characteristics, offset jets are a prototypical flow configuration to study the physics of complex flow observed in many environmental and engineering applications. For instance, offset jets are relevant in predicting the nature of aircraft exhausts and loading effects of aircraft exhausts on ground structures, in the entrainment and mixing process in gas turbine and boiler combustion chambers, and in energy dissipation devices downstream of hydraulic structures. The offset jet is known to combine the characteristics of free, impingement and wall jets, and it is relatively more complex compared to these types of flows. In fact, based on its flow structure, a turbulent offset-jet can be described by three main regions. Close to the nozzle exit, an offset jet has properties similar to those of a free jet. Then, the entrainment of fluid between the jet, the offset wall and the bottom wall creates a low pressure zone, forcing the jet to deflect towards the wall and eventually attaches to it at the impingement point. This is referred to as the Coanda effect [START_REF] Tritton | Physical Fluid Dynamics[END_REF]. Furthermore, downstream after the reattachment point, the offset jet has the characteristics of a wall jet flow.

Many parameters can influence the behavior of an offset jet flow. In earlier studies [START_REF] Bourque | Reattachment of a two-dimensional incompressible jet to an adjacent flat plate[END_REF][START_REF] Sawyer | The flow due to a two dimensional jet issuing parallel to a flat plate[END_REF][START_REF] Mcree | The effect of aspect ratio and offset ratio on nozzle flow and jet reattachment[END_REF][START_REF] Rajaratnam | Plane turbulent reattached wall jets[END_REF][START_REF] Hoch | Theoretical and experimental temperature distribution in two dimensional turbulent jet boundary interaction[END_REF][START_REF] Pelfrey | Mean flow characteristics of a turbulent offset jet[END_REF][START_REF] Nasr | A turbulent plane offset jet with small offset ratio[END_REF], the reattachment length has usually been the primary parameter of interest to study the characteristics of an offset jet flow.

Previous measurements were made with different techniques such as hotwires, Pitot tubes and Laser Doppler Anemometry (LDA). These studies are mainly concerned with the static pressure distribution and velocities measurements in determining the reattachment length. They developed a more general uniform equation that leads to predict the reattachment length as a function of the offset ratio (d/w). With the technology advancement and the recent discovery of coherent structures in the flow, more researchers shifted their interests to studying these turbulent structures. Wang and Tan [START_REF] Wang | Experimental investigation of the interaction between a plane wall jet and a parallel offset jet[END_REF] presented an experimental investigation of a dual-jet formed by a plane wall jet and a parallel offset jet using the Particle Image Velocimetry (PIV) method. They considered an offset ratio of 2 with a Reynolds number equals to 10 4 based on the single jet width relating to an initial jet velocity of 1 m/s. Dynamic characteristics of the flow, including the Reynolds stresses have been discussed.

They observed that the near field of the flow is characterized by a periodic large-scale Karman-like vortex shedding similar to what would be expected in the wake of a bluff body.

Other investigations focused on the effect of the offset ratio (the ratio of the distance from the jet exit to the impingement bottom wall and the jet nozzle diameter). Detailed velocity measurements of 3D offset jets are carried out by Davis and Winarto [START_REF] Davis | Jet diffusion from a circular nozzle above a solid plane[END_REF]. The measurements of mean velocities and turbulence intensities are performed by means of the hotwires technique at a Reynolds number Re=170.000 and with four offset ratios (d/h =0.5, 1, 2 and 4). The authors reported measurements only beyond the reattachment point (x > x mp ). They found that the displacement of the nozzle from the boundary wall effects the position of the attachment point (x mp ). The most comprehensive study of 3D offset jets with various offset ratios was reported by Agelin-chaab and Tachie [START_REF] Agelin-Chaab | Characteristics and structure of 3D offset jets[END_REF][START_REF] Agelin-Chaab | Characteristics of turbulent three-dimensional offset jets[END_REF]. The authors examined the effects of the offset ratio and Reynolds number on the structure of the turbulent round offset jet using the Particle Image Velocimetry (PIV) at three different Reynolds numbers of 5000, 10,000 and 20,000. They found that the decay and spread rates are nearly independent of the Reynolds number and the offset height.

Nyantekyi-Kwakye et al. [START_REF] Baafour | Flow characteristics within the recirculation region of three-dimensional turbulent offset jet[END_REF] presented experimentally the flow characteristics within the recirculation region of the three-dimensional offset jets using a Particle Image Velocimetry technique. Measurements are performed for a rectangular offset jet with four nozzle offset height ratios of 0, 2, 4 and 8. Analysis of the flow field showed that the maximum mean velocity decay rate together with the reattachment length of the jet increased with the increase of the offset height ratio within the recirculation region. The heat transfer characteristics of turbulent offset jet flows are investigated in many papers [START_REF] Heung | Flow and heat transfer characteristics of a two dimensional oblique wall attaching offset jet[END_REF][START_REF] Vishnuvardhanarao | Computation of mean flow and thermal characteristics of incompressible turbulent offset jet flows[END_REF][START_REF] Kanna | Heat transfer study of two-dimensional laminar incompressible offset jet flows[END_REF][START_REF] Kumar | Mean flow characteristics of a turbulent dual jet consisting of a plane wall jet and a parallel offset jet[END_REF]. The effect of different parameters (offset ratio, Prandtl number, Reynolds number and Grashof number) is examined to investigate the flow behavior of conjugate heat transfer. The results are presented in terms of local Nusselt number, local heat flux, wall temperature, and total heat transfer.

Recently, Assoudi et al. [START_REF] Assoudi | Experimental and numerical study of an offset jet with different velocity and offset ratios[END_REF] derived an experimental and numerical study of a turbulent 3D offset jet discharged into a co-flow stream with different offset and velocity ratios. Measurements data of mean velocities and turbulence characteristics are reported using the Particle Image Velocimetry (PIV). A numerical simulation of the problem is also carried out with the second order turbulent Reynolds Stress Model (RSM). Three velocity ratios R = u 0 /u ∞ , (R = ∞, R = 1 and R > 1), and two offset heights (h/d#=#5.3 and 7.8) are investigated to describe the dynamic and turbulent characteristics of the flow. They observed that the increase of the offset ratio gives a better distribution of the jet within the flow field, giving rise to a better dynamic mixture.

Most previous studies are conducted for different values of Reynolds number and offset ratios considered to be of the same fluid property. From this point, the aim of this present paper is to investigate the behavior of a three-dimensional turbulent offset jet injected with different densities issuing into the ambient air at various offset heights (h/w), using a Laser Doppler Velocimetry (LDV) technique (TSI). The specific application for an offset jet with variable densities is to control the rate of the water particles (through the variation of the jet densities) in order to find solutions that promote the sale of fresh products and fight against food waste. This study is specifically intended to provide experimental data for the development and validation of numerical models. The numerical three-dimensional model is simulated through the resolution of the different governing Navier-Stokes' equations by means of the finite volume method. The effect of the density ratio (ρ j /ρ ∞ ) as well as the offset height ratio (h/w) and the distance between the nozzle ejection and impingement surface on the flow structure of 3D offset jets is examined.

Experimental facility and measurement method

Experimental facility

The experimental measurements are conducted in a test section of rectangular dimensions 1500 mm, 400 mm, and 400 mm in the longitudinal, vertical and lateral directions, respectively. The side walls were made of smooth acrylic to facilitate the optical access. Fig. 1 shows a sketch of the test section of the set up experiment and the cartesian coordinate system used in this study [START_REF] Amina | wind tunnel experiments of multijets in crossflow: effect of the injection ratio[END_REF]. The jet nozzle is of a rectangular section of length L=60mm and width w=6mm and offset by a height h above the boundary wall. Note that x=0 is at the$offset$ wall$containing$the$jet exit, y=0 is on the surface wall level, and z=0 is the jet symmetry.

LDV system and measurement procedure

In the current study, two method groups are used to investigate the flow structure, namely, the Laser Tomography System employed for the flow visualization and the Laser Doppler Velocimetry (LDV) technique (TSI) performed for the velocity measurements. The jet is formed by an air flow seeded with small particles of water cloud of approximately 3 µm diameter (Nebulization, ARECO®). The Laser tomography is a technique of instantaneous viewing used in analyzing the structure of turbulent flows. It consists of an illumination system (Nd-YAG, 120 mJ/pulse laser of 532 nm wavelength) utilized to illuminate the flow field and a digital camera (Power View 4 M high resolution cross-correlation camera) with a speed of 1280 x1024 pixels CCD used to image the flow field (Fig. 1) . The velocity measurements were carried out by a non-intrusive LDV technique at a point (a small, non-intrusive optical probe volume) defined by the intersection of two laser beams. This technique is based upon a TSI Power View system, including a 120-mJ dual Nd YAG laser, which produces two flat pulses, like PIV system [START_REF] Saïd Nejla | Experimental and numerical analysis of pollutant dispersion from a chimney[END_REF][START_REF] Amina | Twin inclined jets in crossflow: Experimental investigation of different flow regimes and jet elevations[END_REF]. It also produces of a transmission optics (Bragg cell, beam expanders, beam splitter, prisms, and focusing lens), receiving optics, signal processor units, seeding generation and computer with a data acquisition board and data handling software called "FLOWSIZER". As shown in Fig. 2 the basic elements of the LDV technique are presented; the Laser Doppler Velocimeter sends a monochromatic laser beam toward the target and collects the reflected radiation. According to the Doppler Effect, the change in wavelength of the reflected radiation is a function of the targeted object's relative velocity. Thus, the velocity of the object can be obtained by measuring the change in the wavelength of the reflected laser light, which is done by forming an interference fringe pattern. The absolute error of the position is very small = 0.1 mm (the LDV probe displacement is done by two optical benches).

Test conditions

The experiments were performed for a rectangular nozzle with an offset height h = 33w. The Reynolds number, which is defined as the ratio of the inertial forces to the viscous forces, is based on the jet exit velocity

Computational setup

Governing equations

Consideration is given to a steady, three-dimensional, incompressible and turbulent flow. The equations describing this flow are obtained in a system of Cartesian coordinates with the origin located at the intersection of three plans: the vertical plan containing the jet exit (x=0), the ground wall level (y=0) and the symmetry plane (z=0). The choice of the cartesian coordinates system is motivated by the asymmetry of the jet distribution within the domain in spite of the symmetry of the configuration. Once discretized in the cartesian coordinate system, the equation system is written as follows [START_REF] Baouab | Assessment of a chimney jet flowing around an obstacle[END_REF]:

( ) i i u 0 x ∂ ρ = ∂ % !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (1) ( ) ( ) j i '' '' i i j ij j i j j u u u p u u - g x x x x ∞ ∂ ρ $ % ∂ ∂ ∂ = - + µ -ρ + ρ ρ δ ( ) ( ) ∂ ∂ ∂ ∂ * + % % % !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (2) ( ) j t j j f j u f f x x Sc x ∂ ρ # $ % & µ ∂ µ ∂ = + ' ( ) * ∂ ∂ σ ∂ ' ( , - . / % % % !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (4)! Where, D S c ν = is the Schmidt number.
In the RANS equations, it appears fluctuating functions and variable terms, such as the Reynolds stress tensor (

' ' ' ' j i u u ρ -
), which are required to be modeled. Therefore, the resolution of this system of equations requires a turbulent closure model able to properly characterize the fluctuating functions. In order to choose the appropriate closure model whose results reproduce the experiments better, the k-ε standard turbulent model and the RSM second-order model are compared. The k-ε standard model is the most widely used for the modeling of turbulent flows. It is the simplest one because it solves two separate transport equations [START_REF] Mahjoub Saïd | Experimental and numerical modelling of the three-dimensional incompressible flow behaviour in the near wake of circular cylinders[END_REF].

The Reynolds-Stress Model (RSM) attempts to solve the transport equations for the Reynolds stresses, together with an equation for the dissipation rate. This implies the introduction of several transport equations for all the Reynolds stress components. The RSM model presents the advantage of detailing the three-dimensional aspect of complex flows by computing the destruction of the turbulence through the use of the dissipation rate of the turbulent kinetic energy. The introduction of the RSM model leads to solve the following equation [START_REF] Saïd Nejla | Three-dimensional numerical calculations of a jet in an external cross flow: application of dispersion of pollutants[END_REF]:

j '' '' '' '' '' '' '' '' T i k i j i j i k j k ij ij ij ij k k k k k u u ( u u u ) (u u ) u u u u D G x x x x x ∂ " # ∂ ∂ ∂ ∂ ρ = µ -ρ + + + +φ + ε ( ) ∂ ∂ ∂ ∂ ∂ * + % % % (7) C ij denotes the convective term and L ij D , P ij , T ij D , G ij , ij ϕ , ij
ε , represent respectively, the molecular diffusion, the stress production, the turbulent diffusion, the buoyancy production, the pressure strain and the dissipation rate [START_REF] Ben Kalifa | Fluid flow phenomena in metals processing operations: Numerical description of the fluid flow field by an impinging gas jet on a liquid surface[END_REF]. The equation of the turbulent kinetic energy (k) and that of the dissipation rate of the kinetic energy (ε) which are relative to the second-order model are defined as follows [START_REF] Kalifa Rim | Effect of nozzle-to-plate spacing on the development of a plane jet impinging on a heated plate[END_REF]:

C ij P ij L ij D ! ( ) ( ) ( ) j t ii ii j j k j u k k 1 P G x x x 2 ∂ ρ # $ µ + µ ∂ ∂ = + + -ρ ε ' ( ∂ ∂ σ ∂ ' ( * + % !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (8) ( ) ( ) 2 j t 1 i i 2 j j j u 1 C P C x x x 2 k k ε ε ε ∂ ρ ε $ % µ + µ ∂ ∂ε ε ρε = + - ' ( ∂ ∂ σ ∂ ' ( * + % !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (9) ! 3.

Flow configuration and boundary conditions

The computational configuration is summarized in Fig. 3. The jet emitted from a rectangular nozzle with a variable-density discharging into quiescent ambient surroundings with a density ρ∞=1.2. The nozzle is placed at a bottom distance h=200 mm (33$w),$and discharges with a velocity U j . As illustrated in Fig. 3, the size of the computational domain adopted in this study is 250 w × 66 w × 66 w. The Reynolds number based on the hydraulic diameter jet and the exit velocity is in the range of 2000 to 2900.

A velocity inlet boundary condition is imposed on the jet exit (corresponding to x =0). Various uniform velocities U j and jet density flows ρ j were ejected, with a turbulent intensity I=5.3% calculated experimentally.

Jets with various densities are obtained by changing the density of the air flow at the nozzle exit. The ideal gas law is applicable and the temperature of the jet T j is imposed initially by the following relationship:

273.15 1.292

j j T ρ =
. Wall functions assumptions are imposed to the impingement wall with zero roughness. The use of this approach (wall function) requires that the first grid point adjacent to the wall is within the logarithmic region: 30< y + <100. At the outlet, along the x and z directions, the boundary zero gradients in the normal direction are prescribed.

Numerical method

The resolution of the above governing equations with appropriate boundary conditions is performed with a numerical code, using the Finite Volume Method (FVM). This method allows the discretization of these equations using the first-order upwind scheme [START_REF] Amamou | Computational study of mass and heat transport in a counterflowing turbulent round jet[END_REF]. The SIMPLE algorithm [START_REF] Assoudi | Comparative study of flow characteristics of a single offset jet and a turbulent dual jet[END_REF] is applied for the pressurevelocity coupling. The equations are solved iteratively using the Tri-Diagonal-Matrix-Algorithm (TDMA). The solution is supposed convergent when the residual value of each variable is less than 10 -5 .

The resolution of the last equations system requires a very fine meshing in a great part of the domain. In order to describe exactly any dynamic or shear stress variations, particularly near and immediately downstream the jet, a non-uniform grid was adopted in this study. Furthermore, a sufficiently fine grid is used near the neighboring region of the jet and near the walls (the jet bottom wall and the offset wall) where a very high gradient of variables prevailed in the viscous sub-layer. A three-dimensional grid with 180 × 92 × 76 was elaborated for the computations, which was picked out after a careful grid independence investigation.

Results and discussion

Flow visualization

Fig. 4 represents the visualization of the flow progression under the influence of the jet density variation (ρ j = 1.2, 1.25, 1.3, and 1.4). This figure is actually a sequence of images captured in a series of CCD camera shots on the symmetry plane (z = 0) under the offset ratio of h=33w. It highlights the possibility of free progression of the jet before bending under the influence of the coanda effect. This free progression depends on several factors such as the offset height, the injection ratio and the jet density at the nozzle exit. Referring to Fig. 4, in the case of lighter gases, when the jet density is decreased, the jet is more likely to impose their own behavior which elapses horizontally parallel to the impingement wall before being deflected to the bottom wall.

Indeed, this region corresponds to the potential core which is characterized by a similar turbulent mixing as in the free jet flows. Downstream of the nozzle exit, on the outer edge of the jet, it is observed the formation of the ring-like vortices also known as Shear Layer Vortices (SLV). The formation of these vortices is caused by the presence of the Kelvin-Helmholtz instabilities. This phenomenon appears when the flow is subjected to shearing between two fluids which slip one over the other at different velocities. This structure leads to the formation of a "swirling sheet" within the mixing zone when the flow starts to curve towards the impingement surface. Herein, it is noted that the jet increase in width and the fluid is subjected to the effects of the adverse pressure gradient and the strong interaction with the boundary layer. Therefore, the centerline velocity decreases rapidly as it approaches the wall, and the pressure increases inside the jet until it reaches a maximum value at the attachment point. Within the impingement point, part of the inner shear layer fluid is deflected upstream from the attachment point into the recirculation region. This latter consists of an important characteristic; the reattachment length (x mp ) which is relevant in determining the development of the flow in this region. The reattachment length was defined as the streamwise distance from the nozzle exit to the point where the jet reattached to the bottom wall.

Therefore, the attachment point was estimated as the location on the offset plate above where the reverse flow stops and further downstream, the flow becomes positive. It can be seen that the jet reattaches to the wall at

x mp /d h = 45, 38, 31.8 and 28.2 for ρ j = 1.2, 1.25, 1.3 and 1.4, respectively. It is observed that the reattachment length is basically dependent on the jet density, and it decreases with the increase of the jet density. Further downstream in the lighter jet (ρ j = 1.2 and 1.25), larger structures appear at the upper border of the jet. For the heavier jet these structures seem to have a larger size and the distance between the vortex rings appears to be larger than that observed in the lighter gases.

In Fig. 5, a sequence of images captured under the symmetry plane (z=0) in the case of jet density ρ j = 1.3 and for different offset ratios (h/w = 8, 16, 25 and 33) is shown. In all the injection cases, the jet emerges and penetrates through the quiescent medium immediately downstream of the nozzle exit. Therefore, the jet is deflected, rolled up and skewed, resulting in the formation of jet Shear Layer Vortices (SLV), which are the result of the Kelvin-Helmholtz instabilities of the annular shear layer that separates from the edge of the jet nozzle. The effect of the offset height is clearly observed in the near-field, which is indicated by the breakdown of the flow after a short distance downstream of the nozzle exit. This breakdown eventually leads to the attachment of the jet to the bottom wall in the impingement point. This latter is clearly seen to increase with the offset height ratio (h/w). The impingement point values (x mp /d h ), obtained from the average of the four approaches, for h/w = 8, 16, 25 and 33 are 9, 11.8, 18.1 and 22.7, respectively. A closer look at the jet in the near-field region shows that the flow is influenced by the variation of the offset ratio. In fact, the recirculation region is greatly reduced when the injected jet location approaches to the impingement plate. Consequently, it is noted a reduction of the size of the upright vortices developed in this region between the jet, the offset wall and the bottom wall.

The mean streamlines and the iso-contours of the mean velocity are shown in Fig. 6, in the symmetry plane (z =0) for the different jet densities (ρ j =1.2, 1.25, 1.3 and 1.4) and for an offset height h=33 w. The streamlines are plotted to illustrate the main qualitative features of the offset jet flow in the recirculation and development regions. It is clearly shown that the jet entrains an ambient fluid above and below it. From Fig. 6, the jet spreads similarly for all jet density cases. The jet flow maintains its initial velocity value along a particular downstream distance till the potential core is consumed. After that, the jet deflects toward the bottom wall and the velocity decreases with the downstream distance until it reaches zero at the attachment point. Also, it is worth noting that there is a line which takes place beneath the jet and separates the recirculation region within which the streamwise mean velocity is negative. This line is known as 'the bifurcation line'. Furthermore, it is observed that the bifurcation line decreases in length as the jet density increases. This decrease may be explained by the larger recirculation zone in the case of lighter jets. On the other hand, Fig. 6 shows a backflow region characterized by the existence of negative streamwise mean velocities. In the two first cases where the jet density is ρ j =1.2 and 1.25, no sustained recirculation is present. However, the recirculation zone is clearly observed in the two other gas jets (ρ j =1.3 and 1.4) indicated by the reverse of the streamlines upstream the impingement point. This is due to the short reattachment length in these jets, allowing to a short size of the recirculation zone. These observations reflect previously mentioned deductions about the relation between the jet density and the velocity decay.

Computational code validation

Fig. 7 shows the radial evolution of the streamwise mean velocity (U/U m ) for a gas jet with ρ j = 1.3, at selected locations as indicated in the figure. These locations are picked out carefully in order to describe the mean flow of the offset jet within the three flow regions; the recirculation region (Fig. 7.a), the reattachment region (Fig. 7. b) and wall jet region (Fig. 7.c). When located within the first region (Fig. 7.a), it is obvious that negative values of mean velocity exist close to the wall in the inner region. This is an indication that the ambient fluid entrains from the wall region into the jet. Besides, it is found that U profiles do not collapse entirely within the recirculation and the development regions (Fig. 7.a, b). This can be attributed to the effect of the jet curvature towards the bottom wall and the strong interaction between the inner and the outer shear layers in the impingement region. Nevertheless, U profiles show a fairly collapse in the outer region. Further downstream, in the wall jet region (Fig. 7.c), axial velocity attains self-similarity at all streamwise locations. Also, it is worth noting that the location of the local maximum value of U shifts downwards from y m = 18.18d h at x/d h = 5 in the recirculation region to a constant value of y m = 0.9d h in the wall jet region (Fig. 7.c).

In the following,$ a comparison between experimental data and computational results, carried out by numerical simulation, is reported in Fig. 8 and9. The K-ε Standard model and the second order RSM model are tested in order to find out which closure model reproduces better the current configuration. Fig. 8 represents the variation of the normalized centerline velocity (U c /U j ) according to the streamwise locations (x/d h ). A satisfying agreement between numerical and experimental results is obtained for the four studied jet densities (ρ j = 1.2, ρ j = 1.25, ρ j = 1.3 and ρ j = 1.4). A little discrepancy between RSM and K-ε models is observed in Fig. 8, especially for the three first air densities (ρ j = 1.2, 1.25 and 1.3). This discrepancy is located in the last flow region showing that the second order RSM model predicts better the jet evolution. $ Fig. 9 features the evolution of the normalized longitudinal velocity component (U/U m ) for ρ j = 1.3, at different streamwise positions along the three regions of the flow. Along downstream locations, U profiles are characterized by a single velocity peak. This peak is shifted close to the bottom wall by going further downstream of nozzle exit (y m = 0.9d h at x/d h = 55). Furthermore, the curves did not collapse entirely with the experimental data within the two first zones (recirculation and reattachment regions) whereas a good agreement is noticed beyond the wall jet region (x/d h = 32 and x/d h = 55). This is actually due to the much more turbulent character in the reverse flow zone and the transition of the flow from the impingement region to the wall jet region. Likewise, a good agreement between predicted results obtained with the RSM model and the experimental data is found, as for the centerline velocity decay (Fig. 9). So, the second order closure model RSM is adopted in numerical simulation for the rest of the paper.

Effect of density variation

The development of the mean flow is examined by representing the decay of the centerline velocity (U c ) normalized by the jet velocity (U j ) with the normalized downstream distance x/d h , for various jet densities and for an offset ratio h/w = 33. Fig. 10, mainly demonstrated the dependence of the centerline velocity decay on the density variation. In fact, immediately downstream of the jet nozzle, there is no interesting difference on the velocity profiles; they collapse reasonably well for all the jet densities. This is explained by the existence of the potential core region in which the velocity along the centerline axis remains constant. Nevertheless, the discrepancy between U c /U j curves is clearly observed from the region x/d h ≥ 20, where the centerline velocity decay is more rapid for the denser jet than that for lighter gases. At x/d h = 48, for example, U c /U j ≈ 0.168, 0.108, 0.12 and 0, respectively, for ρ j =1.2, ρ j =1.25, 1.3 and ρ j =1.4, indicating that a lighter jet decays more slowly than a heavy one. This is due to the quicker mixing of the heavy jet with the ambient air. This remark is comforted by the fact that "heavy gas tends to mix more rapidly with the ambient air than the lighter gases. This faster mixing of the heavier gas is accompanied by a faster increase of turbulence intensity in the near region (see Fig. 10).

The streamwise mean velocities at different selected locations of the domain are shown in Fig. 11, for different jet densities and for an offset ratio h/w = 33. The definition of these locations is extracted in order to characterize the mean flow within the different regions of the offset jet. At the first position (x/d h = 9), located in the reveres flow region, it is clear that the mean velocity distribution is independent of the jet density. In fact, the axial velocity curves collapse reasonably well in this region for all available jet densities. It is also obvious that the velocity profile adopts a Gaussian distribution with a single peak quasi-symmetric relative to the jet-axis.

Downstream of the gas injection, the maximum mean velocity decreases with the streamwise distance owing to the deviation of the jet flow towards the wall by the entrainment of the ambient fluid on the jet. Furthermore, y m moved closer to the bottom wall with increasing the jet density; the heavier is the jet, the faster is its drop. The influence of the jet density appears while moving downstream (at x/d h =28 and x/d h =45). Indeed, U profiles do not collapse anymore in the impingement region and in the wall jet region.

For a fixed offset height (h= 33w) and for various gas jet densities, Fig. 12 represents the profiles of the turbulence quantities " "u u , " "v v and " "v u at two streamwise locations; x/dh=9 (in the near region) and x/dh=45 (in the far field region). At the first station (x/dh=9), the density variation seems to have a limited effect on the different Reynolds stresses' distributions. The profiles of these quantities are consequently brought together till attaining self-similarity in this region. Nevertheless, the impact of the density variation appears downstream within the wall jet region (x/d h ≥45) through the increase of the registered peak by increasing the density of considered gases. The maximum level of the " "u u stress is stronger than the " "v v component; they have obtained peak values of 0.14 and 0.112 respectively for the denser jet. This implies that the increase in the jet densities allows the increase of the peaks attained by the shear stress component.

Effect of offset ratios

Fig. 13 shows the streamwise mean velocity U, at selected x/d h locations for different offset ratios using y 0.5 (the jet half width) as the appropriate length scale for the generic offset jet. Immediately downstream of the jet nozzle, for the locations x/d h = 2, 4, 7 and 9, there is no interesting difference on the mean velocity profiles: they are similar regardless of the offset ratio differences. Indeed, these profiles depicted the same global behavior in the different x/d h locations: a Gaussian velocity profile that progressively adopts a peak reached at the position y = 0.87y 0.5 . It is obvious that the jet flow in these positions located at the first region and this peak correspond to the end of the potential core. Far downstream of the jet nozzle, for x/d h = 45 and 55, we observed that the location of the local maximum velocities decrease with increasing h/w and the location of the peak moves downwards in the wall jet region to a constant value y = 0.25y 0.5 for h/w = 8 (Fig. 13.a). This is in accordance with values reported in previous observations (Agelin chaab and Tachie, 2011).

The streamwise development of the offset jet flows was characterized using the decay of the maximum streamwise velocity (U m ) and the wall-normal location of maximum velocity y m (Fig. 14). Previous results as well as those in Davis and Winarto [START_REF] Davis | Jet diffusion from a circular nozzle above a solid plane[END_REF] and Chaab and Tachie [START_REF] Agelin-Chaab | Characteristics and structure of 3D offset jets[END_REF] are considered for comparison. It is obvious, from Fig. 14.a, within the early region of the flow development (x/d h ≤ 36), that there is no interesting difference on the U m profiles: they are similar regardless of the offset heights variation. In fact, the U m values collapsed reasonably well for all the offset ratios near the nozzle exit. One can also see, from this examination, higher initial decay of the flow close to the nozzle. This indicates a constant U m within the potential core region for different offset ratios. This observation is consistent with previous measurement results of U m values. Further downstream in the region 40≤x/d h ≤80, the local maximum velocity (U m ) decays monotonically with the streamwise distance, and one can note a little change on the plotted U m profiles under the different tested offset heights' difference. The development of the location (y m ) with the streamwise distances (x/d h ) for various jets shown in Fig. 14.b, used to exhibit the effects of the offset ratio on the growth of the inner shear layer. It should be noted that the variation of the y m profiles can occur into three regions: The near region (x/d h ≤20), which is characterized by the existence of the potential core where y m remains a constant value. Then, a sharply decrease in the y m value is clearly identified in the second region (x/d h ≤80 in the case of h/w = 33). Finally, we note a monotonic increase of y m in the third region, and that's clearly shown in Fig. 14.b (h/w =8). This indicates that the transition from the second region to the third region is strongly dependent on the offset ratios variation.

The effect of offset height on the Reynolds stresses' distribution are shown in Fig. 15, for the jet density ρ j =1.3 and within various offset ratios (h/w = 8, 16, 25, and 33). It comes from this examination that the more distinct stress peaks are depicted in the region close to the jet exit; which does not automatically mean higher stress values. Within the first location (x/d h = 9) for example, we observed that " "u u and " "v v profiles present double peaks under the lowest offset height which are engendered by the approaching of the jet flow to the impingement wall. Further downstream, at x/d h = 45, the two shear stress components attain peaks whose magnitudes are comparable for all the last three offset ratios. Moving to the " "v u component and within the first jet location, the profiles are going to be reversed, two opposite peaks whose negative magnitudes increase in the vicinity of the bottom wall as we move away from the injection plate. This characterizes the momentum transfer flow between the inner shear layer and the recirculation flow region.

Conclusion

The dynamic evolution of three dimensional offset jets under the effect of the density variation and offset ratio on their turbulent behavior is investigated in this paper through experimental and numerical tests. A confrontation between simulated flows and experimental measurements is done using two tested turbulence models, leading to opt for the second order RSM model. Obtained results revealed the presence of a complex resulting flow field. This complexity is shown by the establishment of an elaborated vortical system composed of two main vortices: the recirculation zone vortex structures and the shear layer vortices. In the near jet exit region, the development of the flow is strongly dependent on the jet density and the offset height. The reattachment length is basically dependent on the jet density; it decreases with the increase of the jet density but increases with the offset height. In the early region, no interesting effect of the variation of the density and the offset ratio was observed on the decay of the maximum mean velocities. However, beyond x/d h ≥ 20, the decay of the maximum mean velocities was observed to be more affected by the variation in these two parameters. Results also revealed that the analysis of the Reynolds stresses' components are nearly independent of the density variation and the offset height in the near region. Nevertheless, the effect of these parameters is clearly observed in the development region. Indeed, a faster mixing of the heavier gas is accompanied by a faster increase of turbulence.

Figure Captions List

( 3 and 1 . 4 .

 314 U j ) and the jet nozzle Hydraulic diameter (d h ) as Re j (U j d h /v) = 2000, 2300, 2600 and 2900. Four different jet density flows ρ j are tested in the outlet ρ j = 1.2, 1.25, 1.3 and 1.4. The jet density variation is obtained by changing the water aerosol concentration entrained by the air flow (ARECO®: Air Refreshing Control). Detailed measurements were taken at various locations downstream of the nozzle in the three regions: the reverse flow, impingement and wall jet regions[START_REF] Assoudi | Three-dimensional study of turbulent flow characteristics of an offset plane jet with variable density[END_REF]. Two series of a plane offset jet experiments were conducted. The first series was done for the nozzle with an offset height h = 33w and for different jet density flows: ρ j = 1.2, 1.25, 1.The second series of experiments is conducted the jet density ρ j = 1.3 and for different offset heights: h = 8w, 16w, 25w and 33w.

  Experimental data are obtained by means of the LDV technique and numerical simulations are performed using the finite volume method together with a turbulent closure model. LDV measurements are conducted for four different Reynolds numbers (based on the jet exit velocity and hydraulic diameter): 2000, 2300, 2600 and 2900.

Fig. 1 .

 1 Fig. 1. Experimental setup.

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. LDV technique measurements Fig. 3. Geometric configuration of the offset jet Fig. 4. Evolution of the flow field in the symmetry plane (z = 0) near the jet exit for different densities.

Fig. 5 .

 5 Fig. 5. Flow visualization in the symmetry plane near the jet exit region for different offset ratios.

Fig. 6 .

 6 Fig. 6. Mean streamlines and iso-contours of (a-d) mean velocities for different density gases: (a) ρ j = 1.2, (b) ρ j = 1.25, (c) ρ j = 1.3 and (d) ρ j = 1.4 (h/w = 33).

Fig. 7 .

 7 Fig. 7. . Profiles of streamwise mean velocities, U, of the offset jet for gas density ρj = 1.3.

Fig. 8 .

 8 Fig. 8. Mean streamwise velocity along the jet axis for different density jet.

Fig. 9 .

 9 Fig. 9. Profiles of the streamwise mean velocities at different selected positions for density jet ρ j =1.3.

Fig. 10 .

 10 Fig. 10. Centerline velocity decay for different density jet.

Fig. 11 .

 11 Fig. 11. Streamwise mean velocity distribution of the offset jets for different jet densities at different longitudinal locations of the domain.

Fig. 12 .

 12 Fig. 12. Comparison of the Reynolds stress distribution for variable densities jet for offset ratio h/w = 33.

Fig. 13 .

 13 Fig. 13. Streamwise mean velocity distribution of the offset jets for different offset ratios: (a) h/w = 8, (b) h/w = 16, (c) h/w = 25 and (d) h/w = 33 (ρ j = 1.3).

Fig. 14 .

 14 Fig. 14. Variation of (a) local maximum mean velocity U m , (b) location of U m and y m within different offset ratios (ρ j = 1.3).

Fig. 15 .

 15 Fig. 15. Comparison of the Reynolds stress distribution for various offset ratios for density jet ρ j = 1.3.

Fig. 1 .Fig. 2 .Fig. 3 .!Fig. 4 .

 1234 Fig. 1. Experimental setup!

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .

 567891011121314 Fig. 6. Mean streamlines and iso-contours of (a-d) mean velocities for different density gases: (a) ρ j = 1.2, (b) ρ j = 1.25, (c) ρ j = 1.3 and (d) ρ j = 1.4 (h/w = 33).

Fig. 15 .

 15 Comparison of the Reynolds stress distribution for various offset ratios for density jet ρ j = 1.3.

ACKNOWLEDGEMENTS

The authors would like to express their gratitude to Faculty of Science, King Khalid University, Saudi Arabia for providing administrative and technical support.


Radhakrishnan Nair
email: nair@liv.ac.uk

Jean-Louis Verger-Gaugry ✸
email: louis.verger-gaugry@univ-smb.fr

Michel Weber
email: michel.weber@math.unistra.fr

I Suriajaya

ON GOOD UNIVERSALITY AND THE RIEMANN HYPOTHESIS

Keywords: Ergodic Averages, Boole's Transformation, Dedekind Zeta Function, Dirichlet L-function, L-Series, Hurwitz Zeta Function, Riemann Zeta Function, The Lindelöf Hypothesis, The Riemann Hypothesis. 2020 Mathematics Subject Classification:

We use subsequence and moving average ergodic theorems applied to Boole's transformation and its variants and their invariant measures on the real line to give new characterisations of the Lindelöf Hypothesis and the Riemann Hypothesis. These ideas are then used to study the value distribution of Dirichlet L -functions, and the zeta functions of Dedekind, Hurwitz and Riemann and their derivatives. This builds on earlier work of R. L.

INTRODUCTION

Equivalent statements of the Riemann Hypothesis are well-known and can be found e.g. in Titchmarsh [T] or more recently e.g. in Mazur and Stein [MSt]. The Lindelöf Hypothesis asserts that for every ε > 0 we have

(1.1) |ζ (1/2 + it)| = O(|t| ε ) as |t| → ∞,
where ζ (z) is the Riemann zeta funtion, and Landau's notation "O" means that there exists a neighbourhood V of ∞ and a constant c ≥ 0 such that: |ζ (1/2 + it)| ≤ c|t| ε for all t ∈ V . Before stating the main Theorems 1.1-1.5, and Theorem 7.2 in the context of moving average ergodic theorems, let us make precise the context in probability theory.

Let (X i ) i≥1 be a sequence of independent Cauchy random variables, with characteristic function φ (t) = e |t| and consider the partial sums S n = X 1 + . . . + X n (n = 1, 2, . . .).

M. Lifshits and M. Weber [START_REF] Lifshits | Sampling the Lindelöf Hypothesis with the Cauchy random walk[END_REF] studied the value distribution of the Riemann zeta function ζ (s) sampled along the random walk (S n ) n≥1 showing, for b > 2, that

(1.2) lim N→∞ 1 N N ∑ n=1 ζ 1 2 + iS n = 1 + o (log N) b N 1 2
.

They also showed that

(1.3) sup n≥1 | ∑ n q=1 ζ ( 1 2 + iS q ) -n| n 1 2 (log n) b 2 < ∞.
Here of course f (x) = o(g(x)) means lim x→∞ f (x) g(x) = 0. This result was extended to Lfunctions and Hurwitz zeta functions by T. Srichan [Sr]. In [START_REF] Steuding | Sampling the Lindelöf Hypothesis with an ergodic transformation[END_REF] J. Steuding replaced (S n ) n≥1 in [START_REF] Lifshits | Sampling the Lindelöf Hypothesis with the Cauchy random walk[END_REF] by (T n x) n≥1 for almost all x on R with respect the Lebesgue measure, for the Boolean dynamical system by T x := x -1

x . This result has its roots in the observation, due to G. Boole [Bo ], subsequently developed by J.W.L. Glashier [G1] [G2], that if f is integrable on the real numbers, then

∞ -∞ f (x)dx = ∞ -∞ f x - 1 x dx.
See [AdW] for a proof of the ergodicity of this dynamical system. In Theorem 3.1 in [LS], the maps

(1.4) T α,β (x) = α 2 x+β α -α x-β if x = β , β if x = β ,
for α > 0 and real β , are shown to be measure preserving and ergodic with respect to the probability measure

(1.5) µ α,β (A) = α π A dt α 2 + (t -β ) 2 ,
for any Lebesgue measureable subset A of the real numbers.

As noted in [LS], if λ denotes Lebesgue measure then µ α,β (A) ≤ 1 απ λ (A) for all A ∈ B, where B denotes the Lebesgue σ -algebra. Also if φ α,β (x) = αx + β , with T = T 1,0 and µ = µ 1,0 we have (1.6)

T α,β = φ α,β • T • φ -1 α,β , which implies the µ α,β -measure preservation and ergodicity of T α,β . The referee of this paper has brought to the authors' attention the paper [MSr], which considers a family of rational maps more general than that in [LS], which we consider in this paper. The suggestion is that our methods in this paper will adapt to the more general framework in [MSr] -at least to a degree. We will not however explore this possibility in this paper, as it would be too much of a digression.

We consider a number of general definitions. We say (k n ) n≥1 ∈ N is L p -good universal if for each dynamical system (X , β , µ, T ) and for each g ∈ L p (X , β , µ) the limit

(1.7) ℓ T,g (x) = lim N→∞ 1 N N-1 ∑ n=0 g(T k n x)
exists µ almost everywhere. Also throughout the rest of this paper B will be the Lebesgue σ -algebra on R and when a σ -algebra on R is not explicitly mentioned it is in fact B.

We say that a sequence x 1 , . . ., x N , . . . is uniformly distributed modulo one if

(1.8) lim N→∞ 1 N #{1 ≤ n ≤ N : {x n } ∈ I} = |I|
for every interval I ⊆ [0, 1). For a real number y we have used {y} to denote its fractional part and let [y] = y -{y} denote its integer part. We say a sequence of integers is uniformly distributed on Z if it is uniformly distributed among the residue classes modulo m, for each natural number m > 1. We say a sequence of natural numbers (k n ) n≥1 is Hartman uniformly distributed (on Z) if it is uniformly distributed on Z, and for each irrational number α, the sequence ({k n α}) n≥1 is uniformly distributed modulo one . This condition coincides with (k n ) n≥1 being uniformly distributed on the Bohr compactification of Z. Some basics on Hartman uniform distribution can be found in the book of Kuipers and Niederreiter [KN].

Note that if (k n ) n≥1 is Hartman uniformly distributed on Z, and if for z with |z| = 1 we set (1.9) F (N, z)

:= 1 N N-1 ∑ n=0 z k n (N = 1, 2, • • •) then (1.10) lim N→∞ F(N, z) = 1 if z = 1, 0 otherwise.
For a c ∈ R we use H c to denote the half plane {z ∈ C : ℜ(z) > c} and use L c to denote the line {z ∈ C : ℜ(z) = c}. We establish the following theorem.

Theorem 1.1. Let f be a meromorphic function on H c satisfying the following conditions:

(1) there exists a K > 0 and a c ′ > c such that for any t ∈ R, we have

(1.11) | f ({σ + it|σ > c ′ })| ≤ K;
(2) there exists a non-increasing ν : (c, ∞) → R such that if σ is sufficiently near c then ν(σ ) ≤ 1 + c -σ and that for any ε > 0, we have f (σ + it) ≪ f ,ε |t| ν(σ )+ε as |t| → ∞;

(3) f has at most one pole of order m in H c at s 0 = σ 0 + it 0 , that is, we can write its Laurent expansion near s = s 0 as

a -m (s -s 0 ) m + a -(m-1) (s -s 0 ) (m-1) + . . . a -1 (s -s 0 ) -1 + a 0 + ∞ ∑ n=1 a n (s -s 0 ) n for m ≥ 0 where we set m = 0 if f has no pole in H c .
Then if (k n ) n≥1 is L p -good universal and Hartman uniformly distributed, for any s ∈ H c \L σ 0 , we have

(1.12) lim N→∞ 1 N N-1 ∑ n=0 f (s + iT k n α,β (x)) = α π R f (s + iτ) α 2 + (τ -β ) 2 dτ,
for almost all x in R.

In the case k n = n (n = 1, 2, . . .) Theorem 1.1 appears in [LS], where it is shown, using (1), ( 2) and (3) and contour integration, for

l α,β (s) := α π R f (s + iτ) α 2 + (τ -β ) 2 dτ, that if f has a pole at s 0 = σ 0 + it 0 , (1.13) l α,β (s) =        f (s + α + iβ ) + B m (s o ), if c < ℜ(s) < σ o , s = s o -α -iβ , m ∑ n=0 a -n (-2α) n , if c < ℜ(s) < σ o , s = s o -α -iβ , f (s + α + iβ ), if ℜ(s) > σ o , or l α,β (s) = f (s -α + iβ ) if s ∈ H c
and f has no pole. Here ℜ(s) denotes the real part of s and

B m (s 0 ) = m ∑ n=1 a -n i n (β + iα -i(s -s 0 )) n - a -n i n (β -iα -i(s -s 0 )) n .
Moreover, when m = 1, we can extend Theorem 1.1 to the line L σ 0 by setting

l α,β (σ 0 + it) = f (σ 0 + α + i(t + β )) - a -1 α α 2 + (t 0 -t -β ) 2 .
Applications will be given in the next section.

As noted earlier Steuding [START_REF] Steuding | Sampling the Lindelöf Hypothesis with an ergodic transformation[END_REF] showed that if ζ denotes the Riemann zeta function

(1.14) lim N→∞ 1 N N-1 ∑ n=0 ζ (s + iT n x) = 1 π R ζ (s + iτ) 1 + τ 2 dτ
exists µ almost everywhere (for x). We can measure the stability of these averages another way. Henceforth C, possibly with subscripts, will denote a positive constant, though nonnecessarily the same on each occasion.

Theorem 1.2. Suppose s = σ + it with 1 2 < σ < 1. Let Y N (σ ) = Y N (σ , x) := 1 N N-1 ∑ n=0 ζ (s + iT n x). (N = 1, 2, . . .)
Suppose (N k ) k≥1 is a strictly increasing sequence of positive integers. Then there exists an absolute C > 0 such that

(1.15) ∞ ∑ k=1 sup N k ≤N<N k+1 |Y N (s) -Y N k (s)| 2 2 ≤ C πσ ζ (2σ ) 2 + ζ (2σ -1) 2σ -1 .
There are other ergodic theoretic measures of the stability of the Riemann zeta function. In this direction we consider the frequencies associated to the sequences

ζ (s + iT n x) n n≥1 . Evidently lim n→∞ ζ (s + iT n x) n = 0,
for almost all x. It is possible to make this more precise as follows. Suppose 0 < p < ∞.

For a sequence of real numbers

x = {x n : n ≥ 1} set (1.16) x p,∞ := sup t>0 {t p #{n : |x n | > t}} 1 p . Theorem 1.3. (i) Suppose 1 2 < σ < 1. Then (1.17) lim m→∞ #{n : |ζ (σ +iT n x)| n ≥ 1 m } m = ζ (σ + 1) - 2 σ (2 -σ ) ,
for almost all x ∈ R. and in L 1 (µ) norm.

(ii) Further

(1.18) |ζ (σ + iT n x)| n : n ≥ 1 1,∞ 1 < ∞. Moreover, Theorem 1.4. Suppose s = σ + it with 1 2 < σ < 1. Suppose κ = (k n ) n≥1 is L p -good universal for p > 2 and let (1.19) R N (s, x, κ) := 1 N N-1 ∑ n=0 ζ (s + iT k n x). (N = 1, 2, . . .).
Then (i) there exists a constant C > 0 such that

(1.20) ∞ ∑ N=1 R N+1 (s, ., κ) -R N (s, ., κ) 2 2 ≤ C πσ ζ (2σ ) 2 + ζ (2σ -1) 2σ -1 , (ii) there exists C > 0 (1.21) µ α,β x : ∞ ∑ N=1 |R N+1 (s, x, κ) -R N (s, x, κ)| 2 ≥ λ ≤ C λ 2 ζ (2σ ) 2 + ζ (2σ -1) 2σ -1 .
We will provide additional information about various special families of sequences in Section 4 and Section 8. On the other hand if we consider the L 1 -norm we have the following theorem in the opposite direction.

Theorem 1.5. Suppose κ = (k n ) n≥1 is Hartman uniformly distributed and L p -good universal for fixed p ≥ 1 and σ ∈ ( 1 2 , 1). Then (1.22)

∑ N≥1 R N+1 (σ , x, κ) -R N (σ , x, κ) = +∞,
almost everywhere in x with respect to Lebesgue measure.

It would be interesting to get good bounds almost everywhere in x for

J ∑ N=1 R N+1 (σ , x, κ) -R N (σ , x, κ) , (J = 1, 2 . . .)
even for specific cases of κ.

The paper is organized as follows: in Section 2 new characterisations of the (extended) Lindelöf Hypothesis are obtained for the Riemann zeta function, and for Dirichlet Lfunctions, Dedekind zeta functions of number fields, Hurwitz zeta functions, as applications of Theorem 1.1. The sequences here are assumed to be Hartman uniformly distributed and L p -good universal. Replacing this assumption by being Stoltz leads to similar limit theorems, which are reported in Section 7. The role played by sublinear sequences having controlled growth is investigated in Section 8 for the Riemann zeta function. Section 6 contains some examples of L p -good universal sequences and Hartman uniformly distributed sequences. Comparison between the dynamical and probabilistic models of Weber [W], resp. Srichan [Sr], is done in Section 5.

APPLICATIONS OF THEOREM 1.1

From now, we assume that (k n ) n≥1 satisfies the hypothesis in Theorem 1.1. In the sequel, for a function f denote f (0) = f and f (k) the k-th derivative of f and set

P k (s) := (-1) k k! i k+1 1 (β + iα -i(s -1)) k+1 - 1 (β -iα -i(s -1)) k+1
for any non-negative integer k.

2.1. The Riemann zeta function.

Theorem 2.1. Suppose (k n ) n≥1 is L p -good universal for some p ∈ [1, ∞] and Hartman uniformly distributed. Then for any k ≥ 0 and s ∈ H -1 2 \L 1 we have

(2.1) lim N→∞ 1 N N-1 ∑ n=0 ζ (k) (s + iT k n α,β (x)) = α π R ζ (k) (s + iτ) α 2 + (τ -β ) 2 dτ
for almost all x in R.

Denote the right hand side of this limit by l

(k) α,β . Then (2.2) l (k) α,β =      ζ (k) (s + α + iβ ) + P k (s), if -1 2 < ℜ(s) < 1, s = 1 -α -iβ , (-1) k γ k -k! (2α) k+1 , if -1 2 < ℜ(s) < 1, s = 1 -α -iβ , ζ (k) (s + α + iβ ), if ℜ(s) > 1,
where

(2.3) γ k = lim N→∞ N ∑ n=1 log k n n - log k+1 N k + 1 .
In the case k = 0 we can extend the result to the line L 1 by setting

(2.4) l (0) α,β = l (0) α,β (1 + it) = ζ (0) (1 + α + i(t + β )) - α α 2 + (t 2 + β 2 )
.

If we set k n = n (n = 1, 2, . . .) Theorem 2.1 appears in [LS] and this is in the case α = 1, β = 0 and k = 1 in [START_REF] Steuding | Sampling the Lindelöf Hypothesis with an ergodic transformation[END_REF].

We also mention here that the case k n = n (n = 1, 2, . . .) and T = T 1,0 of Theorems 2.3 and 2.4 to follow, appeared in [START_REF] Steuding | Sampling the Lindelöf Hypothesis with an ergodic transformation[END_REF], of Theorem 2.4 appeared first in [BSY] and of Theorems 2.6 and 2.7 appeared in [LS].

Proof. Note that for k ≥ 0, we know that ζ (k) is meromorphic and has an absolutely con- vergent Dirichlet series for ℜ(s) > 1. Thus the condition (1) of Theorem 1.1 is satisfied for c ′ > 1. This is because the Laurent expansion of ζ near the pole at s = 1 is known [Br] and then that the Laurent expansion of ζ (k) has the form

ζ (k) (s) = (-1) k k! (s -1) k+1 + (-1) k γ k + ∞ ∑ n=k (-1) n+1 γ n (n -k + 1)! (s -1) n-k+1 .
Thus if k ≥ 0 the function ζ (k) has a pole of order k + 1 at s = 1. We can in addition show (Titchmarsh [T], pp. 95-96) that given ε > 0 we have

ζ (k) (σ + it) ≪ |t| µ(σ )+ε , where µ(σ ) =    0, if σ > 1, (1-σ ) 2 , if 0 ≤ σ ≤ 1, 1 2 -σ , if σ < 0, if k ≥ 0. Therefore Theorem 2.1 follows from Theorem 1.1 with c = -1 2 , s 0 = 1 and m = k + 1 to ζ (k) (s) as required.
We now state a formulation of the Lindelöf Hypothesis in terms of ζ (k) from [LS].

Lemma 2.2. The Lindelöf Hypothesis (1.1) is equivalent, for every ε > 0, to

(2.5) ζ (k) ( 1 2 + it) ≪ |t| ε
as |t| tends to ∞, for any k ≥ 0.

We now give a new characterization of the Lindelöf Hypothesis for the Riemann zeta function, which generalizes the classical one. The classical case k n = n (n = 0, 1, 2, . . .) appears in [LS].

Theorem 2.3. Suppose (k n ) n≥1 is L p -good universal for some p ∈ [1, ∞] and Hartman uniformly distributed. Suppose k is any non-negative integer. Then the statement, for any natural number l,

(2.6) lim N→∞ 1 N N-1 ∑ n=0 ζ (k) 1 2 + iT k n α,β (x) 2l = α π R |ζ (k) ( 1 2 + iτ)| 2l α 2 + (τ -β ) 2 dτ
for µ α,β -almost all x in R, is equivalent to the Lindelöf Hypothesis.

Proof. Via Lemma 2.2 the Lindelöf Hypothesis implies that given ε > 0 we have

ζ (k) 1 2 + . 2m ∈ L p (R, B, µ α,β )
for each pair of natural numbers k, m. Here . represents the variable t ∈ R. Theorem 1.1 and the ergodicity of T α,β implies that

lim N→∞ 1 N N-1 ∑ n=0 ζ (k) 1 2 + iT k n α,β (x) 2l = α π R |ζ (k) ( 1 2 + iτ)| 2l α 2 + (τ -β ) 2 dτ.
We now prove the converse. First

ζ (k) (s) = (-1) k-1 ∞ 1 [x] -x + 1 2 x s+1 (log x) k-1 (-s log x + k)dx + (-1) k k! (s -1) k+1 .
Thus there exists C k > 0, dependent only on k, α and β , such that for |t| ≥ 1 we have

|ζ (k) ( 1 2 + it)| < C k |t|.
Also evidently there exists

c α,β > 0 such that if |τ| ≥ 1 1 α 2 + (τ -β ) 2 ≥ c α,β 1 1 + τ 2 .
Assuming that the Lindelöf Hypothesis is false, implies there exists η > 0 and τ m → ∞ and

C 1 α,β such that ζ (m) ( 1 2 + it) > C 1 α,β τ η m . Now, |ζ (k) ( 1 2 + it)| < C k |t| for any |t| ≥ 1 with C k > 0, and ζ (k) ( 1 2 + iτ) -ζ ( 1 2 + iτ m ) = τ τ m |ζ (k+1) ( 1 2 + it)dt < C 2 α,β |τ -τ m |τ. So |ζ ( 1 2 + iτ)| ≥ 1 2 C 1 α,β τ η m for τ with |τ -τ m | ≤ τ -1 m with m large enough. Let L := 2 3 τ m ; then the interval I := (τ m -τ -1 m , τ m + τ -1 m ) contains the interval (L, 2L) for large m. Hence 2L L ζ (k) 1 2 + iτ 2l 1 2 dτ 1 + τ 2 ≥ C 1 2 2l I τ 2lη-2 m dτ = 2. C 1 2 2l .τ 2lη-3 m ,
which is ≫ T 2lη-3 , and this is impossible as l → ∞. So our theorem is proved.

We now specialize to the case T = T 1,0 and give a condition in terms of ergodic averages equivalent to the Riemann Hypothesis. We denote by ρ = β + iγ, a representative complex zeros of the Riemann zeta function. See Titschmarsh [T], p. 30 for instance for details.

Theorem 2.4. Suppose (k n ) n≥1 is both Hartman uniformly distributed and L p -good universal for some p > 1. Then for almost all x in R with respect to Lebesgue measure we have

(2.7) lim N→∞ 1 N N ∑ n=1 log ζ 1 2 + 1 2 iT k n x = ∑ ρ:ℜ(ρ)> 1 2 log ρ 1 -ρ .
Evidently, the Riemann Hypothesis follows if either side is zero.

To prove Theorem 2.4, we need the following lemma due to M. Balazard, E. Saias and M. Yor [BSY].

Lemma 2.5. We have

(2.8) 1 2π ℜ(s)= 1 2 log |ζ (s)| |s| 2 |ds| = ∑ ρ:ℜ(ρ)> 1 2 log ρ 1 -ρ .
Proof. We wish to use Theorem 1.1 to deduce Theorem 2.4 using Lemma 2.5. To show Theorem 1.1 is relevant we need to show that

log ζ 1 2 + i. ∈ L p (R, µ 1,0 ), (where . denotes t ∈ R), i.e. that ℜ(s)= 1 2 | log |ζ (s))| p | |s| 2 |ds| < ∞.
We mentioned earlier that there exists

C > 0 such that if |t| > 1, ζ 1 2 + it ≤ C|t|.
Also notice that

(log |ζ (s)|) p |s| 2 is continuous on an interval on ℜ(s) = 1 2 centred on s = 1 2 . Away from that interval on ℜ(s) = 1 2 we use the observation that ζ ( 1 2 + it) ≤ C|t|. Hence (for s = 1 2 + it), given δ > 0 we have (log |ζ (s)|) p |s| 2 ≪ 1 |s| 2-δ . This means log ζ 1 2 + i. ∈ L p (R, µ 1,0 ),
as required. Using the fact that T preserves the measure µ 1,0 and is ergodic with respect to this measure, we have

lim N→∞ 1 N N ∑ n=1 log ζ 1 2 + 1 2 iT k n x = 1 2π ℜ(s)> 1 2 log |ζ (s)| |s| 2 |ds|.
µ 1,0 almost everywhere. Theorem 2.4 now follows from Lemma 2.5.

Dirichlet L-functions.

Theorem 2.6. Let L(s, χ) denote the L-functions associated to the character χ. Suppose (k n ) n≥1 is Hartman uniformly distributed and L p -good universal for some p > 1.

Then, for k ≥ 1, (i) if χ is non-principal, for s ∈ H -1 2 we have lim N→∞ 1 N N-1 ∑ n=0 L (k) (s + iT k n α,β (x), χ) = α π R L (k) (s + iτ, χ) α 2 + (τ -β ) 2 dτ (2.9) = L (k) (s + α + iβ , χ) for almost all x in R, (ii) if χ(= χ 0 ) is principal, for s ∈ H -1 2 \L 1 we have (2.10) lim N→∞ 1 N N-1 ∑ n=0 L (k) (s + iT k n α,β (x), χ) = α π R L (k) (s + iτ, χ) α 2 + (τ -β ) 2 dτ,
for almost all x in R.

Denote this limit, i.e. the right hand side of (2.10), by l

(k) α,β (s, χ 0 ). Then (2.11) l (k) α,β (s, χ 0 ) =      L (k) (s + α + iβ , χ 0 ) + γ -1 (χ 0 )P k (s), if -1 2 < ℜ(s) < 1, s = 1 -α -iβ , γ k (χ 0 ) -k!γ -1 (χ 0 ) (2α) k+1 , if -1 2 < ℜ(s) < 1, s = 1 -α -iβ , L (k) (s + α + iβ , χ 0 ), if ℜ(s) > 1,
where γ -1 (χ 0 ), γ k (γ 0 ), are constants that depend on χ 0 . These are the coefficients of the Laurent expansion of L (k) (s, χ 0 ) near s = 1. If k = 0 we can extend the result to the line L 1 by defining

l (0) α,β = l (0) α,β (1 + it, χ 0 ) = L(1 + α + i(t + β ), χ 0 ) - αγ -1 (χ 0 ) α 2 + (t 2 + β 2 )
.

Proof. We know that L (k) (s, χ) has a Dirichlet series expansion for ℜ(s) > 1, for each non-negative integer k. From this we can show that

L (k) (s, χ) ≪ k , ε |t| µ(σ )+ε where µ(σ ) =    0, if σ > 1, (1-σ ) 2 , if 0 ≤ σ ≤ 1, 1 2 -σ , if σ < 0. If χ is non-principal then L (k) (s, χ) is entire for all k ≥ 0, so L (k) (s, χ) satisfies Theorem 1.1 for all s ∈ H -1 2 . If χ = χ 0 is principal, L (k) (s, χ 0 ) (k ≥ 1) has a pole of order k + 1 at s = 1.
We can therefore apply Theorem 1.1 with c = -1 2 , s 0 = 1 and m = k + 1, to L (k) (s, χ 0 ) with Laurent coefficients coming from [IK].

2.3. The Dedekind zeta function of a number field. We now consider the Dedekind ζ K (s) function of a number field K of degree d K over the rationals Q which is defined by as follows. For s ∈ C such that ℜ(s) > 1, let

ζ K (s) = ∑ I⊂O K 1 (N K/Q (I)) s .
Here I runs over the ideals contained in the ring of integers of K denoted O K and N K/Q (I) denotes the absolute norm of I in K, which is the cardinality of the quotient O K /I. In the case K = Q, the function ζ K (s) reduces to the Riemann zeta function ζ (s). The complex function ζ K (s) can be extended meromorphically to the entire complex plane with a simple pole at s = 1. See [Coh] p. 216, for more details.

Theorem 2.7. Suppose (k n ) n≥1 is L p -good universal for some p ∈ [1, ∞] and Hartman uniformly distributed. For any k ≥ 0 and any s ∈ H1

2 -1 d K \L 1 we have (2.12) lim N→∞ 1 N N-1 ∑ n=0 ζ (k) K (s + iT k n α,β (x)) = α π R ζ (k) K (s + iτ) α 2 + (τ -β ) 2 dτ for almost all x in R.
Denote the right hand side of this limit by l

(k) K α,β
. We have (2.13)

l (k) K α,β (s) =      ζ (k) K (s + α + iβ ) + γ -1 (K)P k (s), if 1 2 -1 d K < ℜ(s) < 1, s = 1 -α -iβ , k!γ k (K) - k!γ -1 (K) (2α) k+1 , if 1 2 -1 d K < ℜ(s) < 1, s = 1 -α -iβ , ζ (k) (s + α + iβ ), if ℜ(s) > 1.
Here γ -1 (K) and γ k (K) are constants dependent only on K, which are coefficients of the Laurent expansion of ζ (k)

K near s = 1. Also in the case k = 0 we can extend the result to the line L 1 by setting

(2.14) l (0) K α,β (1 + it) = ζ (0) (1 + α + i(t + β )) - αγ -1 (K) α 2 + (t 2 + β 2 )
.

Proof. We refer to [START_REF] Steuding | On the value-distribution of L-functions[END_REF] for a bound for L on the half line and to [HIKW] for the coefficients of the Laurent expansion ζ K (s) near the pole s = 1. We then proceed as earlier with Theorem 2.1, c = 1 2 -1 d K , s 0 = 1 and m = k + 1. Remark 2.8. Results analogous to the ergodic characterisation of the Lindelöf Hypothesis just given, can be proved for L-functions associated to primitive characters and Dedekind ζ function. In both cases, the statements of ergodic characterisation are of the type: there exist constants k, ε > 0 such that

f 1 2 + it ≪ f ,k,ε |t| ε ,
as |t| → ∞, for suitable L -functions f . This inequality when f is an L-function arising from Dirichlet series is called the Generalized Lindelöf Hypothesis.

2.4. The Hurwitz zeta function. Recall that the Hurwitz zeta function is defined for a > 0 and s ∈ C with ℜ(s) > 1 by

ζ (s, a) = ∞ ∑ n=1 1 (n + a) s .
It is continued meromorphically to the whole of C with a single pole at s = 1.

Theorem 2.9. Suppose (k n ) n≥1 is L p -good universal for some p ∈ [1, ∞] and Hartman uniformly distributed. Then for any s such that ℜ(s) > -1 2 , s = 1, with 0 ≤ a < 1 and k a non-negative integer, we have

(2.15) lim N→∞ 1 N N-1 ∑ n=0 ζ (k) (s + iT k n α,β (x), a) = α π R ζ (k) (s + iτ, a) α 2 + (τ -β ) 2 dτ,
for almost all x in R.

Denote the right hand side of this limit by l

(k) α,β (s, a). Then (2.16) l (k) α,β (s, a) =      ζ (k) (s + α + iβ , a) + P k (s), if -1 2 < ℜ(s) < 1, s = 1 -α -iβ , (-1) k γ k (a) -k! (2α) k+1 , if -1 2 < ℜ(s) < 1, s = 1 -α -iβ , ζ (k) (s + α + iβ , a), if ℜ(s) > 1,
where

(2.17) γ k (a) = lim N→∞ N ∑ n=1 log k (n + a) n + a - log k+1 (N + a) k + 1 .
In the case k = 0 we can extend the result to the line L 1 by setting

(2.18) l (0) α,β (1 + it, a) = ζ (0) (1 + α + i(t + β ), a) - α α 2 + (t 2 + β 2 )
.

If we set k n = n (n = 1, 2, . . .) Theorem 2.1 appears in [LS] and this is the case α = 1, β = 0, and k = 1 in [START_REF] Steuding | Sampling the Lindelöf Hypothesis with an ergodic transformation[END_REF].

Proof. Following the proof of Theorem 2.1 choose c = -1 2 , s 0 = 1 and m = k + 1. For the bound on |ζ (k) (s, a)| on the half line and the coefficients of the Laurent series of ζ (k) (s, a) near s = 1 we refer to [START_REF] Steuding | On the value-distribution of L-functions[END_REF].

PROOF OF THEOREM 1.1

The proof is in the continuation, and a generalization, of a Theorem of [LS] in which conditions (1), ( 2) and (3) are used. We first recall a special case of a Theorem of S. Sawyer [Sa]:

Suppose for a dynamical system (R, B, µ α,β , T α,β ) that g ∈ L p (R, B, µ α,β ) and let g =

( X | f | p dµ α,β ) 1 p . Set Mg(x) = sup N≥1 1 N N ∑ n=1 g(s + iT k n α,β (x)) . (N = 1, 2, . . .) If (k n ) n≥1 is L p -good universal for p > 1, then there exists C > 0 such that (3.1) Mg p ≤ C g p . Because 1 N ∑ N n=1 g(s + iT k n α,β (x)) ≤ Mg(x) (N = 1, 2, . . .)
and Mg ∈ L p , the dominated convergence theorem implies

h(x) = lim N→∞ 1 N N ∑ n=1 h(s + iT k n α,β (x))
exists in L p -norm. Our next order of business is to show that h(s

+ iT α,β (x)) = h(s + ix). Let U α,β g(s + ix) = g(s + iT α,β (x))
. This is a norm preserving operator on L p as T α,β is µ α,β measure preserving. Also let U -1 α,β denote the L 2 adjoint of U α,β . Recall that we say any sequence (c n ) n∈Z is positive definite if given a bi-sequence of complex numbers (z n ) n∈Z , only finitely many of whose terms are non-zero, we have ∑ n,m∈Z c n-m z n z m ≥ 0.

Here z is the conjugate of the complex number z. Let a, b = R abdµ α,β (i.e. the standard inner product on L 2 ). Notice that U n α,β g, g n∈Z is positive definite. Recall that the Bochner-Herglotz theorem [Kt] says that there is a measure ω g on T, such that

U n α,β g, g = T z n dω g (z). (n ∈ Z)
This tells us that

1 N N ∑ n=1 g(s + iT k n +1 α,β (x)) - 1 N N ∑ n=1 g(s + iT k n α,β (x)) 2 = T (2 -z -z -1 ) 1 N N ∑ n=1 z k n 2 dω g (z)
using the parametrization z = e 2πiθ for θ ∈ [0, 1), this is

= 4 T sin 2 θ 2 1 N N ∑ n=1 z k n 2 dω g (z).
Using the fact that sin θ 2 = 0 if θ = 0 and the fact that (k n ) n≥1 is Hartman uniformly distributed, by (1.9) and (1.10), we see that g(s + iT α,β (x)) = g(x). A standard fact from ergodic theory is that if T α,β is ergodic and g(s + iT α,β (x)) = g(x) for measurable g then g(x) is constant, which must be R gdµ α,β . This extends to the L p -norm for all p > 1.

All we have to do now is to show that the pointwise limit is the same as the norm limit, i.e. that g(x) = g(x) = R gdµ α,β . We consider the sequence of natural numbers

(N t ) t≥1 such that 1 N t N t ∑ n=1 g(s + iT k n α,β (x)) - R g(x)dµ α,β p ≤ 1 t . Thus ∞ ∑ t=1 X 1 N t N t ∑ n=1 g(s + iT k n α,β (x)) - R g(x)dµ α,β p dµ < ∞. Fatou's lemma gives R ∞ ∑ t=1 1 N t N t ∑ n=1 g(s + iT k n α,β (x)) - R g(x)dµ α,β p dµ < ∞.
This implies

∞ ∑ t=1 1 N t N t ∑ n=1 g(s + iT k n α,β (x)) - R g(x)dµ α,β p < ∞.
almost everywhere. This means

1 N t N t ∑ n=1 g(s + iT k n α,β (x)) - X g(x)dµ α,β = o(1). µ α,β almost everywhere. As (k n ) n≥1 is L p -good universal we must have lim N→∞ 1 N N-1 ∑ n=0 g(s + iT k n α,β x) = R g(s + ix)dµ α,β
µ α,β almost everywhere as required to prove Theorem 1.1.

4. PROOF OF THEOREMS 1.2 TO 1.5

We begin by recalling the spectral regularization method of Lifshits and Weber [LW1] [LW2], and some technical preliminaries. Assume (X , β , ν) is a measure space with finite measure ν and that S is a ν measure preserving transformation of (X , β , ν).

For f ∈ L 1 (X , β , ν) let (4.1) B N ( f ) = 1 N N-1 ∑ k=0 f • T k . (N = 1, 2, . . .)
We need the following lemma, due to R. Jones, R. Kaufman, J. Rosenblatt and M. Wierdl [JKRW].

Lemma 4.1. Suppose (N p ) p≥ is a strictly increasing sequence of positive integers. Then there exists an absolute constant C > 0 such that

(4.2) ∞ ∑ p=1 sup N p ≤N<N p+1 |B N ( f ) -B N p ( f )| 2 2 ≤ C f 2 .
Write log + (u) = max(1, log(u)) for u ≥ 1. Let ω denote the spectral measure associated to the element f and define its regularised measure ω via its Radon-Nykodim derivative by

d ω dx (x) = π -π Q(θ , x)ω(dθ ),
where

Q(θ , x) = |θ | -1 log 2 + ( θ x ) if |x| < |θ |, θ 2 |x| -3 if |θ | ≤ |x| ≤ π.
The following is a theorem of Lifshits and Weber [START_REF] Lifshits | Spectral regularization inequalities[END_REF].

Lemma 4.2. Suppose N 0 and N 1 with N 0 < N 1 are positive integers. Then there exists an absolute constant C > 0 such that

(4.3) sup N 0 ≤N<N 1 |B N ( f ) -B N 0 ( f )| 2 2 ≤ C ω [ 1 N 1 1 N 0 ) . Suppose 0 < p < ∞.
For a sequence of real numbers x = {x n : n ≥ 1} the quantity x p,∞ is defined in (1.16). Also if r < p we have

x p,∞ ≤ x p ≤ p p -r 1 p x p,∞ .
We also consider

N * x = sup m≥1 #{n : x n n > 1 m } m .
In our considerations, x) . We have the following Lemma due to I. Assani [As] (see also [W] -we refer to Jamison, Orey and Pruitt [JOP] for more on the random variable case, and [RR] for Birnbaum-Orlicz spaces and the "L log + L" notation).

x n = f (τ n (x)) (n = 1, 2, . . .) and x = O f (x) = ( f (τ n (x)) n≥1 which is the value of f along the orbit of a point x ∈ X . Also let N f (x) = N * O f (
Lemma 4.3. (i) For any non-negative f ∈ L 1 (µ) the function N f (x) is weak-(p, p) for all p ∈ (1, ∞). Further (4.4) lim m→∞ #{n : f (τ n (x)) n > 1 m } m = X f dµ,
µ almost everywhere and also in L 1 (µ)-norm.

(ii) For f ∈ L log + L (4.5) f (τ n (x)) n 1,∞ 1 < ∞.
Note that, for two positive constants C 1 ,C 2 ,

C 1 N f (x) ≤ f (τ n (x)) n , n ≥ 1 1,∞ ≤ C 2 N f (x).
We now turn to the proof of Theorem 1.2. By Lemma 4.1,

∞ ∑ k=1 sup N k ≤N<N k+1 |Y N (σ ) -Y N k (σ )| 2 2 ≤ C π R |ζ (σ + iτ)| 2 1 + τ 2 dτ,
where as before C is universal. We note that, for σ ∈ ( 1 2 , 1),

R |ζ (σ + it)| 2 1 + t 2 dt = ∞ 0 {x} x 2σ +1 dx = - 1 σ ζ (2σ ) 2 + ζ (2σ -1) 2σ -1 .
Here again {x} = x -[x] denotes the fractional part of x. The quantity in the bracket on the right hand side is negative. Further,

1 2π R |ζ ( 1 2 + it)| 2 1 4 + t 2 dt = log(2π) -γ.
where γ is Euler's constant. See Prop. 7, Cor. 8 and (1.26) in Coffey [C]. Thus

(4.6) C π R |ζ (σ + iτ)| 2 1 + τ 2 dτ ≤ C πσ ζ (2σ ) 2 + ζ (2σ -1) 2σ -1 .
completing the proof of Theorem 1.2. We now turn to the proof of Theorem 1.3. The first assertion is a consequence of Lemma 4.3 (i). The limit is

1 π R |ζ (σ + it)| 2 1 + τ 2 dt.
This integral is a special case of integrals calculated in [START_REF] Steuding | On the value-distribution of L-functions[END_REF]. As a consequence of Theorem 1.1 in [START_REF] Steuding | Sampling the Lindelöf Hypothesis with an ergodic transformation[END_REF],

1 π R |ζ (σ + it)| 2 1 + τ 2 dt = ζ (σ + 1) - 2 σ (2 -σ ) .
For the second assertion, it follows from Lemma 4.3 (ii). The function ζ (s) ∈ L log + L is integrable, with respect to the Cauchy measure, because of the estimate

ζ (σ + it) ≪ ε t 1-σ 2 +ε .
See Titchmarsh, [T] section 5.1, for the details of this. Therefore Theorem 1.3 is proved as required.

To prove Theorem 1.4 we need the following two lemmas. Suppose (X , β , µ) is a measure space and that T : X → X is a measure preserving map. Let (a k ) ∞ k=0 be a sequence of natural numbers and for any measurable f on X set (4.7)

C N f (x) := 1 N N-1 ∑ k=0 f (T a k x), (N = 1, 2, • • •)
that is the ergodic averages corresponding to the sequence and let

M f (x) := sup N≥1 |C N f (x)|.
Further from the data (X , β , µ, T ) and f we construct the ergodic q-variation function

V q f (x) = ( ∑ N≥1 |C N+1 f (x) -C N f (x)| q ) 1 q . (q ≥ 1)
Our first lemma is Theorem 1 from [START_REF] Nair | On variation functions for subsequence ergodic averages[END_REF].

Lemma 4.4. Suppose for a sequence of natural numbers (a k ) ∞ k=0 that for some p > 1 and C p > 0 depending only on p and (X , β , µ, T ) we have

(4.8) M f p ≤ C p f p .
Then there exists another constant D p > 0 depending only on p and (X , β , µ, T ) such that if q > 1 then (4.9) V q f p ≤ D p f p .

Further suppose there is a constant C > 0 depending only on (X , β , µ, T ) such that we have

(4.10) µ({x : M f (x) > λ }) ≤ C λ X | f | dµ.
Then there is a constant D > 0 depending only on (X , β , µ, T ) such that if q > 1 then

(4.11) µ({x : V q f (x) > λ }) ≤ D λ X | f | dµ.
Proof. Suppose (X , L , η) denote a finite measure space i.e. η(X ) < ∞. Now, let (T n ) n≥1 denote a sequence of linear transformations of L p (X , L , η) into measurable functions on X , such that each

T n is continuous in measure, that is, such that if || f -f m || p tends to 0 as m tends to ∞, then ||T n f -T n f m || p also tends to 0 as m tends to ∞. Set T * f (x) = sup n≥1 |T n f (x)| for f ∈ L p (X , L , η
). We will say the family (T n ) n≥1 commutes with w :

X → X if T * f (w(x)) ≤ T * g(x)
where g(x) = f (w(x)) on X . Now suppose F is a family of ergodic η preserving transformations on X , closed under composition. We will call (T n ) n≥1 distributive if it commutes with all the elements of some family F on X . We have the following result of S. Sawyer [Sa].

Lemma 4.5. Let (T n ) n≥1 be a distributive sequence of linear operators on L p (X , L , η), where each T n is continuous in measure and maps L p (X , L , η) to measurable functions on X . If p ∈ [1, 2] and if T * f (x) < ∞ η almost everywhere, then there exists a uniform constant C > 0 such that (4.12)

η {x : T * f (x) ≥ λ } ≤ C λ p X | f (x)| p dη,
for all λ > 0 and f ∈ L p (X , L , η).

The conclusion of Lemma 4.5 is that if T * f (x) < ∞ η almost everywhere, then the operator T * satisfies a weak * (p, p). If there is a C > 0 such that ||T * f || p ≤ C|| f || p we say T * satisfies a strong (p, p) inequality. It is easy to check that strong (p, p) inequalities imply the corresponding weak (p, p) inequalities for the operator T * . On the other hand it follows from the Marcinkiewicz interpolation theorem [SW] that a weak (p, p) inequality implies a strong (q, q) inequality if q > p. We now specialize (4.1) to the situation where f is defined on X = R, L is the Lebesgue algebra on X = R and η = µ α,β :

T N f (x) := 1 N N ∑ n=1 f (T k n α,β (x)) (n = 1, 2, . . .) with f (x) = ζ (σ + ix).
One checks readily that (T N ) N≥1 commutes with (T n α,β ) n≥1 and is therefore distributive. In light of Lemma 4.5, we see that Theorem 1.4 follows by recalling our estimate

R |ζ (σ + ix)| 2 dµ α,β ≤ C π R |ζ (σ + iτ)| 2 1 + τ 2 dτ ≤ C πσ ζ (2σ ) 2 + ζ (2σ -1)
2σ -1 arising in (4.6) in the Proof of Theorem 1.3, Theorem 1.4 is proved.

It is possible to say more about specific sequences and families of sequences. Suppose N = (N k ) k≥1 and (a n ) n≥1 are sequences of natural numbers. Consider the definition of C N in (4.7). Let

S f (x) = S(N , f )(x) := ∑ k≥1 C N k+1 f (x) -C N k f (x) 2 1 2 .
In the first instance, we are interested in conditions under which there are constants C > 0 such that

(4.13) S f 2 ≤ C f 2 .
We have the following lemma taken from [N2].

Lemma 4.6. Suppose 1 < a ≤ N k+1 N k ≤ b < ∞ and a k = φ (n)
, where φ is a non-constant polynomial mapping the natural numbers to themselves. Then there is a constant C > 0 such that (4.13) holds.

The following lemmas are taken from [START_REF] Nair | On variation functions for subsequence ergodic averages[END_REF].

Lemma 4.7. Suppose 1 < a ≤ N k+1 N k ≤ b < ∞ and a k = φ (p n )
, where φ is as in Lemma 4.6 and p n is the n th rational prime. Then there is a constant C > 0 such that (4.13) holds.

Let θ = (θ k ) k≥1 be a Z valued sequence of independent identically distributed random variables with basic probability space (Ω, B, P). We assume the σ -algebra B is P complete and that there exists γ > 0 such that E (θ + 1 ) γ < ∞. Here we have used θ + 1 to denote max(θ 1 , 0). Consider a strictly increasing sequence (q k ) k≥1 of natural numbers for which S is a bounded map from L 2 to itself when a k = q k and N k = [ρ k ] for some ρ > 0, that is such that

(4.14) ∑ k≥1 C N k+1 ( f ) -C N k ( f ) 2 (x) 1 2 2 ≤ C f 2 .
Also assume there exists δ ∈ (0, 1) such that (4.15)

q k = o(2 k δ ) with (4.16) lim sup k→∞ log q 2k log q k < ∞ and P(q 1 + θ 1 ≥ 0) = 1. Let C θ N ( f ) := 1 N N ∑ k=1 f (T q k +θ k x).
Lemma 4.8. If (q k ) k≥1 satisfies (4.14), (4.15) and (4.16), then there is a set Ω 0 contained in Ω of full P measure such that if ω ∈ Ω 0 there exists a constant C > 0 such that (4.17)

∑ k≥1 C θ N k+1 ( f ) -C θ N k ( f ) 2 (x) 1 2 2 ≤ C f 2 .
Consider two strictly increasing sequences of natural numbers (q k ) k≥1 and N = (N k ) k≥1 such that if a k = q k then (4.14) holds. Let I q = [2 c q , 2 c q+1 ) denote the q th interval of the form [2 a , 2 a+1 ) containing an element of N . Let Φ(N) = (log[q n + q + 2]) 1 2 if N ∈ I q and assume (4.18)

∑ N∈ N Φ 2 (N) N < ∞.
Let φ = (φ k ) k≥1 be a sequence of independent, identically distributed random variables defined on a basic probability space (Ω, β , P) with φ 1 ∈ L 1 (Ω, β , P) such that

(4.19) E sup N∈ N\{0} ∑ N k=1 (φ k -E(φ k )) 2 N 1 2 < ∞. Let w φ N ( f ) := 1 N N ∑ k=1 φ k f (T q k x). (N = 1, 2, • • •)
Lemma 4.9. Suppose (q k ) k≥1 , N = (N k ) k≥1 and φ = (φ k ) k≥1 satisfy (4.16), (4.18) and (4.19). Then for almost all ω in Ω (4.20)

( ∑ k≥1 |w φ k N k+1 ( f ) -w φ k N k ( f )| 2 ) 1 2 2 ≤ C f 2 .
Some remarks about the nature of condition (4.19) are in order.

If N k = [k ε ] (k = 1, 2, . . .), with ε > 1 then (4.19) reduces to ∑ k≥1 log(q [k ε ] + log k) k ε < ∞, which is realised if q k = O(2 k δ ) for δ > 0. Also if N k = 2 k (k = 1, 2, . . .) then (4.19) reduces to ∑ k≥1 log(q 2 k + log k) 2 k < ∞, which is realized if q k = O(2 k γ ).
Given the earlier Lemma of this section these two conditions are not difficult to satisfy.

Set

k n = [g(n)] (n = 1, 2, . . .)
where g is a differentiable function from [0, ∞) to itself whose derivative increases with its argument. Let Z M denote the cardinality of the set {n : k n ≤ M} and suppose for some function a : [1, ∞) → [1, ∞) increasing to infinity as its argument does, that we set

b(M) := sup {α}∈[ 1 a(M) , 1 2 ) ∑ n:k n ≤M e 2πik n α .
Suppose also for some decreasing function c :

(1, ∞) → (0, ∞) that b(M) + Z [a(M)] + M a(M) Z M ≤ C c(M).
Then if we have

∞ ∑ s=1 c(ρ s ) < ∞, for every ρ > 1 we say (k n ) n≥1 satisfies condition H (see examples in Section 6). Let b k = g -1 (k) -g -1 (k -1)
where g -1 here denotes the inverse function of g on its set of definition. Also suppose that there is a constant C such that (4.21)

g -1 ( 1 {α} ) 2 ∑ 1 {α} N k ≥1 1 g -1 (N k ) 2 ≤ C.
Lemma 4.10. Suppose (k n ) n≥1 satisfies condition H, and (4.21) and that 1 < a ≤ In [N] it is shown that examples of sequences of integers which satisfy conditions H include those given by g

(n) = n ω , for non-integer ω > 1, g(n) = e (log n) γ for γ ∈ (1, 3 2 ) and g(n) = α k n k + • • • + α 1 n + α 0 ,
where the real numbers α 1 , • • • , α k are not all multiples of the same real number. The reader will readily verify that these examples also satisfy conditions H and (4. 19). An important point of note is that, as shown in [N], the ergodic averages, for a n = k n and f in L p with p > 1 converge to a T invariant limit.

We want to show that if (k n ) n≥1 satisfies conditions H, (4.19) then (4.13) is satisfied with

1 < a ≤ N k+1 N k ≤ b.
By identifying it with its characteristic function we may view a strictly increasing sequence of natural numbers as a point in the power set of the natural numbers (i.e. 2 N ), or as a point in the Cartesian product Π ∞ n=1 X n where for each natural number n, we have set X n = {0, 1}. As a consequence we may put a probability measure on the space of strictly increasing sequences of integers, as a product measure π by setting π n ({1}) = σ n for σ n ∈ [0, 1] and π n ({0}) = 1 -σ n and defining π to be the Cartesian product measure Π ∞ n=1 π n . For a strictly increasing sequence of integers (N k ) k≥1 suppose also that

(4.22) ∑ k≥1 log 2 N k ∑ n≤N k σ n < ∞,
for arbitrary real α that we have some constant C dependent only on π such that (4.23) 1

∑ n≤ 1 {α} σ n 2 ∑ N k 1 {α} ≥1 1 ∑ n≤N k σ n 2 < C.
Then with regard to the probability measure just defined, we have the following lemmas.

Lemma 4.11. For ergodic averages with regard to almost all strictly increasing sequences (a k ) k≥1 with respect to π, if (N k ) k≥1 and (σ k ) k≥1 satisfy (4.22) and (4.23) above then (4.16) holds.

Lemma 4.12. Fix a natural number d > 2 and let (k n ) n≥1 denote the set (4.24) Λ = {n ∈ N : n = ∑ j≥1 q j d j with q j ∈ {0, 1}},

ordered by size. Then if N k = d k (k = 1, 2, • • •) then (4.13) holds. For s = σ + it set C N (s, ζ , x) := 1 N N ∑ n=1 ζ (σ + iT k n (x)) (N = 1, 2, . . .)
and set

S(s, ζ , x) := ∑ k≥1 C N k+1 (s, ζ , x) -C N k (s, ζ , x) 2 1 2 .
We have the following theorem.

Theorem 4.13.

If σ ∈ ( 1 2 , 1) we have C > 0 such that (4.25) S(s, ζ , .) 2 2 ≤ C πσ ζ (2σ ) 2σ + ζ (2σ -1) 2σ -1 ,
for all the families and sequences (k n ) n≥1 and N listed in Lemmas 4.6 -4.12.

The situation is different if we replace L 2 -norms by L 1 -norms.

Lemma 4.14. Suppose (X , β , µ, T ) is an ergodic and measure preserving transformation with µ non-ergodic. Suppose (k n ) n≥1 is Hartman uniform distributed and L p -good universal for fixed p ≥ 1. Then for any non-constant function f on (X , β , µ) we set

C N ( f ) = 1 N N ∑ n=1 f (T k n (x)) (N = 1, 2, . . .).
Then we have

(4.26) ∑ N≥1 C N+1 ( f ) -C N ( f ) = +∞, µ almost everywhere.
Applying Lemma 4.14 to f (x) = ζ (σ + ix) for σ ∈ ( 1 2 , 1) and x ∈ R we get Theorem 1.5.

COMPARING DYNAMICAL AND PROBABILISTIC

MODELS Let Y = Π ∞ n=1 Ω, that is the space of sequences (X 1 , X 2 , • • •) in Ω. Let p 1 denote the projection p 1 : Y → Ω defined by p 1 ((X 1 , X 2 , • • • )) = X 1 . Also let S : Y → Y denote the shift map defined on Y by S((X 1 , X 2, • • • , )) = (X 2 , X 3 , • • • ). It is routine to check that S preserves the infinite product measure µ ∞ = Π ∞ n=1 µ on Y .
That the shift map T is also ergodic with respect to this infinite product measure is consequence of the Kolmogorov zero one law. Now define f on Y by f (ω 1 , ω 2 , • • • ) = X 1 (ω 1 ). This means that X n (ω) = f (S n-1 y) where y = (ω 1 , ω 2 , • • • ) and S denotes the above shift map. Also a simple computation shows that Y f (y)dµ ∞ = E (X 1 ). This means that the strong law of large numbers follows from an application of Birkhoff's pointwise ergodic theorem and the weak law of large numbers from an application of Von Neumann's norm ergodic theorem respectively.

The upshot of this is that, under quite weak hypotheses, the comparison between the random model described in [W] and the dynamical model in [Sr] is actually a comparison between two different dynamical systems. Now suppose that (k n ) n≥1 is Hartman uniformly distributed and L p good universal for fixed p ∈ [1, ∞) and µ is the Cauchy distribution µ α,β then (5.1) lim

N→∞ 1 N N-1 ∑ n=0 f (s + iX k n (ω)) = α π R f (s + iτ) α 2 + (τ -β ) 2 dτ,
for almost all ω in R. We can specialise this to (5.2) lim

N→∞ 1 N N ∑ n=1 log ζ 1 2 + 1 2 iX k n (ω) = ∑ ρ:ℜ(ρ)> 1 2 log ρ 1 -ρ .
Again, the Riemann Hypothesis follows if either side is zero. As above similar observations can be obtained other zeta functions and L-functions.

The condition of good universality is an assumption about all dynamical systems. We don't need to assume so much and can deduce our conclusions from the properties of one transformation. The following Theorem offers that link between the two models. See also [JO ] where similar ideas appear.

Theorem 5.1. Consider two ergodic dynamical systems (X 1 , β 1 , µ 1 , T 1 ) and (X 2 , β 2 , µ 2 , T 2 ) both on separable measure spaces. Suppose that µ 1 and µ 2 are non-atomic. Then if for a particular sequence of integers (k n ) n≥1 for each f 1 ∈ L p (X 1 , β , µ 1 ) for all p > 1 we have

(5.3) lim N→∞ 1 N N ∑ n=1 f 1 (T k n 1 x 1 ) = X 1 f 1 (x 1 )dµ 1 ,
µ 1 almost everywhere, then the same is true with 1 replaced by 2.

The condition of non-atomicity of µ 1 and µ 2 is not strictly necessary for the proof but it simplifies the proof somewhat and our intended applications are to non-atomic dynamical sytems.

Proof. Let (X , β , µ, T ) denote a dynamical system and for sequence of natural numbers

(k n ) n≥1 let m( f ) = sup N≥1 1 N N ∑ n=1 f (T k n x) .
A special case of a theorem of S. Sawyer [Sa], tells us, after the hypothesis of Theorem 5.1, with m = m 1 , that there exists C ′ p > 0 such that

µ {x 1 ∈ X 1 : m 1 ( f 1 )(x 1 ) ≥ λ } ≤ C p f 1 p λ .
Another way to say this is that the operator m 1 satisfies a weak-(p, p) bound. A stronger assertion is that there exists C p * > 0 such that (5.4) m 1 ( f 1 ) p ≤ C * p f 1 p . Here we say m 1 satisfies a strong-(p, p) inequality. For any fixed p, a strong-(p, p) inequality implies the corresponding weak-(p, p) inequality. On the other hand as a consequence of the Marcinkiewicz interpolation theorem [SW] weak-(p a , p a ) implies strong-(p b , p b ) if p a < p b . This means that the fact that the weak-(p, p) of m for all p > 1 is equivalent to it being strong-(p, p) for all p > 1.

We now show that inequality (5.1) implies (5.5) m 2 ( f 2 ) p ≤ C * p f 2 p Suppose that the dynamical systems (X 1 , β 1 , µ 1 , T 1 ) and (X 2 , β 2 , µ 2 , T 2 ) are both ergodic. Now the argument used to deduce Theorem 1.1 from (3.1) together with the ergodicy (X 2 , β 2 , µ 2 , T 2 ) readily implies the conclusion of Theorem 2.1.

We now prove (5.5). As both the dynamical systems (X 1 , β 1 , µ 1 , T 1 ) and (X 2 , β 2 , µ 2 , T 2 ) are ergodic, the Rokhlin-Halmos lemma says that given any integer N ≥ 1 and ε > 0, there exist sets

E i ⊂ X i (i = 1, 2) with E i , T -1 i E i , . . . , T -(N-1) i E i disjoint and µ i (E i ∪ T -1 i E i . . . ∪ T -(N-1) i E i ) < 1 -ε. Let γ = µ 1 (E 2 ) µ 2 (E 2 )
. Also for a set D and a function f let f D denote the restriction of f to D. Let us observe that there always exists a bijection from E 1 to E 2 that preserves the measure from µ 1 to γµ 2 . This is always possible by the measure isomorphism theorem between separable spaces. Let us call δ such a 1-1 measure map from E 1 to E 2 . It is not canonical. Now set

F N i = E i ∪ T -1 i E i . . . ∪ T -(N-1) i E i , (i = 1, 2)
and extend the definition of

∆| E 1 := δ to F 1 by setting ∆(x) = (T k 2 δ T -k 1 )(x) for x ∈ T k E 1 . Now set W f (y) := f (∆ -1 y)γ 1 p . Notice supp W f (y) ⊂ T k 2 E if supp f (y) ⊂ T k 1 E. Direct computation now gives for supp f (y) ⊂ T k 1 E T k E 1 | f | p dµ 1 = T k E 2 W f p dµ 2 ,
and so

(5.6)

F 1 | f | p dµ 1 = F 2 W f p dµ 2 . Let C i,l ( f )(x) = 1 l l ∑ n=1 f (T k n i x), (l = 1, 2, . . .) and set m i,N ( f )(x) = sup 1≤l≤N |C i,l ( f )(x)|, (N = 1, 2, . . .)
and evidently

m i ( f )(x) = lim N→∞ m i,N ( f )(x).
In proving (5.5) by splitting f into its real and imaginary parts and each of those into their positive and non-negative parts we may assume f ≥ 0. Suppose x ∈ T k 1 E 1 and for the transformation T 1 we have m N ( f )(x) ≥ 0. Then there exists l ∈

[1, N] such that m i,l ( f )(x) = C i,l ( f )(x).
Let y = W x. Then a direct computation shows that for the transformation T 2 we have

C 2,l (W f )(y) = W (m 1,N ( f ))(x).
From this we deduce that m

2,N (W f )(y) ≥ m 1,l ( f )(x). Thus m 2,N ( f ) p p = X 2 m 2,N ( f )(x) p dµ 2 which is ≤ F 2 m 2,N ( f )(x) p dµ 2 + ε.
and we know W fixes measure on F 2 and hence L p norms so we have

F 2 W (m 2,N ( f ))(y) p dµ 2 + ε = (m 2,N (W ( f )))(y) p p dµ 2 + ε ≤ c W ( f ) p p + ε ≤ c f p p d + ε. Now let N → ∞
and let ε → 0 and the proof of the theorem is complete.

The ergodicity of the random dynamical system is implied by the Kolmogorov 0 -1 law. If we choose µ = µ α,β , we see that Theorem 1.1 is equivalent to (5.1). From this we deduce all the applications of Theorems 5.1 with T k n replaced by X k n .

HARTMAN UNIFORMLY DISTRIBUTED AND GOOD UNIVERSAL SEQUENCES

In this section we give some examples of L p -good universal sequences for some p ≥ 1. The examples 1, 3-6 are Hartman uniformly distributed. Example 2 is not Hartman uniformly distributed in general.

1.

-The natural numbers:

The sequence (n) ∞ n=1 is L 1 -good universal. This is Birkhoff's pointwise ergodic theorem.

Let φ be any non-constant polynomial mapping the natural numbers to themselves. Note that if n ∈ N, then n 2 ≡ 3 mod 4, so in general the sequences (φ (n)) ∞ n=1 and (φ (p n )) ∞ n=1 are not Hartman uniformly distributed. We do, however, know that

if γ ∈ R \ Q, then (φ (n)γ) ∞ n=1 and (φ (p n )γ) ∞ n=1 are uniformly distributed modulo 1. 2. Condition H: Sequences (k n ) ∞
n=1 that are both L p -good universal and Hartman uniformly distributed can be constructed as follows. Set

k n = [τ(n)] (n = 1, 2, . . .c), where τ : [1, ∞) → [1, ∞
) is a differentiable function whose derivative increases with its argument. Let Ω m denote the cardinality of the set {n : a n ≤ m}, and suppose, for some function ϕ : [1, ∞) → [1, ∞) increasing to infinity as its argument does, that we set ρ(m) = sup

{z}∈ 1 ϕ(m) , 1 2 ∑ n : k n ≤m e(zk n ) .
Suppose also, for some decreasing function ρ

: [1, ∞) → [1, ∞) and some positive constant ω > 0, that ρ(m) + Ω [ϕ(m)] + m ϕ(m) Ω m ≤ ωρ(m).
probability space (Y, A , P). Assume that there exist 0 < α < 1 and β > 1/α such that a n = O(e n α )E log

β + |θ 1 | < ∞. Then (k n +θ n (ω)) ∞
n=1 is both L p -good universal and Hartman uniformly distributed [START_REF] Nair | On random perturbation of some intersective set[END_REF].

MOVING AVERAGES

In this paragraph we show that the assumption of a (univariate) sequence which has the property to be L p good universal and Hartman uniformly distributed can be removed from the Theorems 2.1, 2.3, 2.4, 2.6, 2.7, 2.9, and replaced by the assumption of a (bivariate) Stoltz sequence.

7.1. Stoltz sequences and Theorem 7.2. Let Z be a collection of points in Z × N and let

Z h := {(n, k) : (n, k) ∈ Z and k ≥ h}, Z h α := {(z, s) ∈ Z 2 : |z -y| < α(s -r) for some (y, r) ∈ Z h } and Z h α (λ ) := {n : (n, λ ) ∈ Z h α } (λ ∈ N).
Geometrically we can think of Z 1 α as the lattice points contained in the union of all solid cones with aperture α and vertex contained in Z 1 = Z. We say a sequence of pairs of natural numbers (n l , k l ) ∞ l=1 is Stoltz if there exists a collection of points Z in Z × N, and a function h = h(t) tending to infinity with t such that (n l , k l ) ∞ l=t ∈ Z h (t) and there exist h 0 , α 0 and A > 0 such that for all integers λ > 0 we have |Z h 0 α 0 (λ )| ≤ Aλ . This technical condition is interesting because of the following theorem [BJR], which will be used in the proof of Theorem 7.2. The Stoltz condition is related to the "cone condition" of Nagel and Stein [NS] and of Sueiro [Su].

Theorem 7.1. Let (X , β , µ, T ) denote a dynamical system, with set X , a σ -algebra of its subsets β , a measure µ defined on the measurable space (X , β ) such that µ(X ) = 1 and a measurable, measure preserving map T : X → X . Suppose g is in L 1 (X , β , µ) and that the sequence of pairs on natural numbers

(n l , k l ) ∞ l=1 is Stoltz. Then (7.1) m g (x) := lim l→∞ 1 k l k l ∑ j=1 g(T n l + j x)
exists almost everywhere with respect to µ.

Theorem 7.2. Let f be a meromorphic function on H c satisfying conditions (1), ( 2) and

(3) of Theorem 1.1. Then if (n l , k l ) l≥1 is Stoltz, for any s ∈ H c \L σ 0 , we have

(7.2) lim l→∞ 1 k l k l ∑ j=1 f (s + iT n l + j α,β (x)) = α π R f (s + iτ) α 2 + (τ -β ) 2 dτ
for almost all x in R. This means that m f (s + iT α,β (x)) = m f (s + i(x)) µ α,β almost everywhere. As an ergodic dynamical system constant along almost all orbits must be constant, we must have m f (x) = R f (s + ix)dµ α,β (x). We have therefore shown that

lim l→∞ 1 k l k l ∑ j=1 f (s + iT n l + j α,β (x)) = α π R f (s + iτ) α 2 + (τ -β ) 2 dτ,
for almost all x in R, as required. 7.2. Applications and moving average ergodic Theorems. We will forgo the proof of the Theorems below as they follow from those of Theorems 2.3, 2.4, 2.6, 2.7, 2.9 and 5.1 respectively via minor modification, in a similar way, using Theorem 7.1 and Theorem 7.2.

Theorem 7.3. Suppose (n q , k q ) q≥1 is Stoltz. Suppose k is any non-negative integer. Then the statement, for any natural number l, (7.5) lim

q→∞ 1 k q k q ∑ j=1 ζ (k) (s + iT n q + j α,β (x)) l = α π R |ζ (k) (s + iτ)| l α 2 + (τ -β ) 2 dτ
for µ α,β -almost all x in R, is equivalent to the Lindelöf Hypothesis.

Theorem 7.4. Suppose (n q , k q ) q≥1 is Stoltz. Then for almost all x in R with respect to Lebesgue measure we have

(7.6) lim q→∞ 1 k q k q ∑ j=1 log ζ ( 1 2 + 1 2 iT n q + j x) = ∑ ρ:ℜ(ρ)> 1 2 log ρ 1 -ρ .
Again, if either side is zero, this is equivalent to the Riemann Hypothesis. We now consider Dirichlet L-functions.

Theorem 7.5. Let L(s, χ) denote the L-series associated to the character χ. Suppose (n q , k q ) q≥1 is Stoltz, and let k be a nonnegative integer. Then, (i) if χ is non-principal, for s ∈ H for almost all x in R.

We now consider the Hurwitz zeta function.

Theorem 7.6. Suppose (n q , k q ) q≥1 is Stoltz. For any s such that ℜ(s) > -1 2 , s = 1, 0 ≤ a < 1 and k a non-negative integer, we have (7.9) lim

q→∞ 1 k q k q ∑ j=1 ζ (k) (s + iT n q + j α,β (x), a) = α π R ζ (k) (s + iτ, a) α 2 + (τ -β ) 2 dτ,
for almost all x in R.

In (7.9) the right hand side is given by (2.16), (2.17) and (2.18). The analogue of Theorem 5.1 is now the following.

Theorem 7.7. Consider two ergodic dynamical systems (X 1 , β 1 , µ 1 , T 1 ) and (X 2 , β 2 , µ 2 , T 2 ), both on separable measure spaces. Suppose that µ 1 and µ 2 are non-atomic. Then if for a Stoltz sequence of integers (n q , k q ) q≥1 and each f 1 ∈ L p (X 1 , β 1 , µ 1 ) we have (7.10) lim q→∞ 1 k q k q ∑ j=1 f 1 (T n q + j 1

x 1 ) = X 1 f 1 (x 1 )dµ 1 , µ 1 almost everywhere, then the same is true with 1 replaced by 2.

SUBLINEARITY AND THE RIEMANN ZETA FUNCTION

In the same vein as Theorem 1.2, Theorem 1.3 and Theorem 1.4, now with peculiar sublinear sequences, the following Theorem can be deduced.

The following Lemma is taken from [START_REF] Lifshits | Régularisation spectrale et propriétés métriques des moyennes mobiles[END_REF]. Let I A denote the indicator function of the set A.

Lemma 8.1. Suppose k q = k(q), q = 1, 2, . . ., with k sub-linear such that sup u k ′ (u) is absolutely bounded. Also assume

K(θ ) = 1 |θ | min( 1 |θ | ,1) k ′ (u) |u| 2 du + k 1 |θ | |θ | I {|θ |≤1} ,
is uniformly bounded on R. Suppose (X , β , µ, T ) is a dynamical system and f ∈ L 2 (X , β , µ) and set

g k n ( f (x)) = 1 n k n +n-1 ∑ j=k n f (T j (x)).
Then, for any strictly increasing sequence of natural numbers (n q ) q≥1 , there exists C > 0 such that

∑ q≥1 |g k n q+1 ( f ) -g k n q ( f )| 2 1 2 2 ≤ C f 2 .
We now specialize to the case, f (x) = ζ (σ + ix) with σ ∈ [ 1 2 , 1). Theorem 8.2. Suppose k and K are as in Lemma 8.1. Now set, with σ ∈ [ 1 2 , 1),

G k n,σ (x) = G k n (ζ (σ + ix)) = 1 n k n +n-1 ∑ j=k n ζ (σ + iT j (x)).
Then, for any strictly increasing sequence of natural numbers (n q ) q≥1 , there exists a constant C > 0 such that

∑ q≥1 G k n q+1 ,σ (ζ ) -G k n q ,σ (ζ ) 2 1 2 2 ≤ C πσ ζ (2σ ) 2 + ζ (2σ -1) 2σ -1 .

  b. Then (4.16) holds for the corresponding square function.

  f (s + iT α,β (x))m l, f (x) = 1 k l ( f (s + iT n l +k l +1 α,β (x))f (s + iT n l +1 α,β (x))).

LL

  (k) (s + iT n q + j α,β (x), χ) = α π R L (k) (s + iτ, χ) α 2 + (τ -β ) 2 dτ (7.7) = L (k) (s + α + iβ , χ) for almost all x in R, (ii) if χ is principal, for s ∈ H -(k) (s + iτ, χ) α 2 + (τ -β ) 2 dτ,

ACKNOWLEDGEMENTS

We thank the referee for very detailed comments, substantially improving the presentation of this paper, and for bringing to our attention important additional references of material significance to this topic.

Then if we have

for all θ > 0, we say that (a n ) ∞ n=1 satisfies condition H, see [N]. Sequences satisfying condition H are known to be both Hartman uniformly distributed and L p -good universal. Specific sequences of integers that satisfy condition H include a n = [τ(n)] (n = 1, 2, . . .c) where:

all rational multiplies of the same real number.

[IV.] Hardy fields: By a Hardy field, we mean a closed subfield (under differentiation) of the ring of germs at +∞ of continuous real-valued functions with addition and multiplication taken to be pointwise. Let H denote the union of all Hardy fields. Conditions for

, where ψ ∈ H to satisfy condition H are given by the hypotheses of Theorems 3.4, 3.5 and 3.8. in [BKQW]. Note the term ergodic is used in this paper in place of the older term Hartman uniformly distributed.

A random example:

Suppose that S = (k n ) ∞ n=1 is a strictly increasing sequence of natural numbers. By identifying S with its characteristic function χ S , we may view it as a point in Λ = {0, 1} N , the set of maps from N to {0, 1}. We may endow Λ with a probability measure by viewing it as a Cartesian product Λ = ∏ ∞ n=1 X n , where, for each natural number n, we have X n = {0, 1} and specify the probability ν n on X n by ν n ({1}) = ω n with 0 ≤ ω n ≤ 1 and ν n ({0}) = 1 -ω n such that lim n→∞ ω n n = ∞. The desired probability measure on Λ is the corresponding product measure ν = ∏ ∞ n=1 ν n . The underlying σ -algebra A is that generated by the cylinders

n=1 in Λ, with respect to the measure ν, is Hartman uniformly distributed (see Proposition 8.2 (i) in Bourgain [Bou ]). Hartman uniformly distributed sequences are called ergodic sequences in [Bou ]. 4. Block sequences:

Suppose that

as n tends to infinity. Note that this allows the possibility that (a n ) ∞ n=1 is zero density. This example is an immediate consequence of Tempelman's semigroup ergodic theorem. See page 218 of [BL]. Being a group average ergodic theorem this pointwise limit must be invariant, which ensures that the block sequence must be Hartman uniformly distributed. 5. Random perturbation of good sequences:

Suppose that (a n ) ∞ n=1 is an L p -good universal sequence which is also Hartman uniformly distributed. Let θ = (θ n ) ∞ n=1 be a sequence of N-valued independent, identically distributed random variables with basic probability space (Y, A , P), and a P-complete σ -field A . Let E denote expectation with respect to the basic