Twisted quadratic moments for Dirichlet $L$-functions
Bull. Korean Math. Soc. 2015 Vol. 52, No. 6, 2095-2105
Published online November 30, 2015
St\'ephane R. Louboutin
163 Avenue de Luminy, Case 907
Abstract : Given $c$, a positive integer, we set $$M(f,c) :={2\over\phi (f)}\sum_{\chi\in X_f^-}\chi (c)\vert L(1,\chi)\vert^2,$$ where $X_f^-$ is the set of the $\phi (f)/2$ odd Dirichlet characters mod $f>2$, with $\gcd (f,c)=1$. We point out several mistakes in recently published papers and we give explicit closed formulas for the $f$'s such that their prime divisors are all equal to $\pm 1$ modulo $c$. As a Corollary, we obtain closed formulas for $M(f,c)$ for $c\in\{1,2,3,4,5, 6,8,10\}$. We also discuss the case of twisted quadratic moments for primitive characters.
Keywords : $L$-function, character, mean values, moments
MSC numbers : Primary 11M20
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