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Number Theory SeminarExport this event to calendar

Tuesday, October 8, 2019 — 1:30 PM EDT

Ertan Elma, Department of Pure Mathematics, University of Waterloo

"Discrete Mean Values of Dirichlet L-functions"

Let χ be a Dirichlet character modulo a prime number p ⩾ 3 and let \mathfrak a_χ:=(1-χ(-1))/2. Define the mean value
\begin{align*}
\mathcal{M}_{p}(s,\chi):=\frac{2}{p-1}\sum_{\substack{\psi \bmod p\\\psi(-1)=-1}}L(1,\psi)L(s,\chi\overline{\psi})
\end{align*}
for a complex number s such that s≠ 1 if \mathfrak a _χ=1.

Mean values of the form above have been considered by several authors when χ is the principal character modulo p and \Re(s)>0 where one can make use of the series representations of the Dirichlet L-functions being considered. In this talk,  we will investigate the behaviour of the mean value \mathcal{M}_{p}(-s,χ) where χ is a nonprincipal Dirichlet character modulo p and \Re(s)>0. Our main result is an exact formula for \mathcal{M}_{p}(-s,χ) which, in particular, shows that
\begin{align*}
\mathcal{M}_{p}(-s,\chi)= L(1-s,\chi)+\mathfrak a_\chi 2p^sL(1,\chi)\zeta(-s)+o(1), \quad (p\rightarrow \infty)
\end{align*}
for fixed 0<σ:=\Re(s)<\frac{1}{2} and  |\Im s|=o\left(p^{\frac{1-2 σ}{3+2 σ}}\right).

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