Bulletin of the
Korean Mathematical Society BKMS

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