Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1035-1048, 2021
On discrete mean values of Dirichlet $L$-functions
Ertan Elma
Received May 12, 2020. Published online February 2, 2021.
Abstract: Let $\chi$ be a nonprincipal Dirichlet character modulo a prime number $p\geq 3$ and let $\mathfrak a_\chi:= \tfrac12 (1-\chi(-1))$. Define the mean value $\mathcal{M}_p(-s,\chi) :=\frac2{p-1} \sum\Sb\psi\pmod p \\ \psi(-1)=-1 \endSb L(1,\psi)L(-s,\chi\bar{\psi})$ $(\sigma:=\Re s > 0).$ We give an identity for $\mathcal{M}_p(-s,\chi)$ which, in particular, shows that $\mathcal{M}_p(-s,\chi)= L(1-s,\chi)+\mathfrak a_\chi2p^s L(1,\chi)\zeta(-s) +o(1)$ $(p\rightarrow\infty)$ for fixed $0 < \sigma < \frac12$ and $|t:=\Im s|=o (p^{(1-2\sigma)/(3+2\sigma)})$.
Keywords: Dirichlet $L$-function; mean value; Dirichlet character
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Affiliations: Ertan Elma, Department of Pure Mathematics, University of Waterloo, 200 University Ave. West, N2L 3G1, Waterloo, ON, Canada, e-mail: eelma@uwaterloo.ca
