## Number Theory Seminar - Ertan Elma (University of Waterloo)

### Monday, February 24th, 2020

**Time:** 4:30-5:30 p.m. **Place:** Jeffery Hall 422

**Speaker:** Ertan Elma (University of Waterloo)

**Title:** Discrete Mean Values of Dirichlet L-functions.

**Abstract:** Let $\chi$ be a Dirichlet character modulo a prime number $p \geq 3$ and let $\mathfrak{a}_{\chi}:=(1-\chi(-1))/2$. Define the mean value $$ \mathcal{M}_{p}(s,\chi):=\frac{2}{p-1}\sum_{\substack{\psi \bmod p\\\psi(-1)=-1}}L(1,\psi)L(s,\chi\overline{\psi})$$ for a complex number $s$ such that $s\neq 1$ if $\mathfrak{a} _{\chi}=1.$ Mean values of the form above have been considered by several authors when $\chi$ 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 $\mathcal{M}_{p}(-s,\chi)$ where $\chi$ is a non-principal Dirichlet character modulo $p$ and $\Re(s)>0$. Our main result is an exact formula for $\mathcal{M}_{p}(-s,\chi)$ which, in particular, shows that $$ \mathcal{M}_{p}(-s,\chi)= L(1-s,\chi)+\mathfrak a_\chi 2p^sL(1,\chi)\zeta(-s)+o(1), \quad (p\rightarrow \infty)$$ for fixed $0<\sigma:=\Re(s)<\frac{1}{2}$ and $|\Im s|=o\left(p^{\frac{1-2\sigma}{3+2\sigma}}\right)$.