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Keywords:
generalized Dirichlet $L$-functions; mean value properties; functional equation; asymptotic formula
Summary:
Let $q\ge 3$ be an integer, let $\chi $ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions $$ L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac {\chi (n)}{(n+a)^s}, $$ where $s=\sigma +{\rm i} t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac 12+{\rm i} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.
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