Zeta FunctionsSeptember 18  22, 2006, Moscow, RussiaLaboratoire J.V. Poncelet 

Yatusaka Ihara Chuo University, Japan Construction and study of "Mfunctions" closely related to the behaviour of logarithmic derivatives of LfunctionsWe shall construct and study two basic functions $M_{\sigma}(z)$ (nonnegative) and (its Fourier transform) $\tilde{M}_{\sigma}(z)$ on the complex plane, each parametrized by $\sigma >1/2$, that are closely related to the behaviour of values of logarithmic derivatives of Lfunctions. They have quite interesting analytic and arithmetic properties. Among them, $\tilde{M}_{\sigma}(z)$ is realanalytic both in $\sigma$ and $z$, and has a convergent Euler product expansion whose pfactor can be expressed in terms of (p and) the Bessel $J_{\nu}$ functions. Both decrease rapidly with $z$ for any fixed $\sigma >1/2$. Fix $s$ with $Re (s)=\sigma$, and let $\chi$ run over all Dirichlet characters with (say) prime conductors. Then (generally conjecturally), $M_{\sigma}(z)$ (resp. $\tilde{M}_{\sigma}(z)$) coincides with the limitaverage over $\chi$ of $\delta_z(L'(\chi,s)/L(\chi,s))$ (resp. $\psi_z(L'(\chi,s)/L(\chi,s))$). Here $\delta_z(w)$ is the Dirac measure concentrated at $z$ and $\psi_z(w)=\exp(i Re(\bar{z}w))$ is the additive character on the field of complex numbers parametrized by $z$. These conjectural statements are in fact valid when $1<\sigma$, and moreover, for a certain function field analogue, they are valid also for some $1/2 < \sigma \leq 1$ (presently, at least when $3/4 <\sigma$). The problem involves Fourier analysis of some arithmetic functions and prime number theory. Go to the Laboratoire Poncelet home page. 

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