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Keywords:
Dirichlet $L$-function; mean value; Dirichlet character
Dirichlet $L$-function; mean value; Dirichlet character
Summary:
Let $\chi $ be a nonprincipal Dirichlet character modulo a prime number $p\geqslant 3$ and let $\mathfrak a_\chi := \tfrac 12 (1-\chi (-1))$. Define the mean value $$ \mathcal {M}_{p}(-s,\chi ) :=\frac {2}{p-1} \sum \psi \pmod p \psi (-1)=-1 L(1,\psi )L(-s,\chi \bar {\psi }) \quad (\sigma :=\Re s>0). $$ We give an identity for $\mathcal {M}_{p}(-s,\chi )$ which, in particular, shows that $$ \mathcal {M}_{p}(-s,\chi )= L(1-s,\chi )+\mathfrak a_\chi 2p^s L(1,\chi )\zeta (-s) +o(1) \quad (p\rightarrow \infty ) $$ for fixed $0<\sigma <\frac {1}{2}$ and $|t:=\Im s|=o (p^{(1-2\sigma )/(3+2\sigma )})$.
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