Zeta Functions

September 18 - 22, 2006, Moscow, Russia

Laboratoire J.-V. Poncelet

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Yatusaka Ihara

Chuo University, Japan

Construction and study of "M-functions" closely related to the behaviour of logarithmic derivatives of L-functions

We shall construct and study two basic functions $M_{\sigma}(z)$ (non-negative) 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 L-functions. They have quite interesting analytic and arithmetic properties. Among them, $\tilde{M}_{\sigma}(z)$ is real-analytic both in $\sigma$ and $z$, and has a convergent Euler product expansion whose p-factor 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 limit-average 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.