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 On the mean values of $L(1,\chi)$ Bull. Korean Math. Soc. 2012 Vol. 49, No. 6, 1303-1310 https://doi.org/10.4134/BKMS.2012.49.6.1303Published online November 30, 2012 Zhaoxia Wu and Wenpeng Zhang Northwest University, Northwest University Abstract : Let $p>2$ be a prime, and let $k\geq 1$ be an integer.Let $\chi$ be a Dirichlet character modulo $p$, and let $L(s,\chi)$ be the Dirichlet $L$-functions corresponding to $\chi$. In this paper we consider the mean values of $$\mathop{\sum_{\chi\bmod p}}_{\chi(-1)=-1}\chi(2^k)\left|L(1,\chi)\right|^2.$$ Keywords : $L$-function, Dirichlet character, identity MSC numbers : 11M06 Downloads: Full-text PDF