On the distribution of the values of $L(1,\chi_{8p})$

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 253-273

Résumé

The moments of pure imaginary and integer orders of the function $L(1,\chi_{8p})$, where $\chi_{8p}(n)=(8p/n)$ and $p$ runs over all primes $p>2$, are computed. In order to derive uniform variants of the theorems on moments, the extended Riemann hypothesis for the Dirichlet $L$-series must be used. As corollaries, the limiting distribution of the values of $\log L(1,\chi_{8p})$ is studied, and quantitative analogs of the $\Omega$-results for $L(1,\chi_{8p})$ are obtained. Previously, $\Omega$-results for $L(1,\chi_p)$ were proved by Bateman, Chowla, and Erdos (1949–1950) and by Barban (1966), and their methods can easily be transferred to $L(1,\chi_{8p})$. Bibliography: 27 titles.