Second moments of Dirichlet $L$-functions weighted by Kloosterman sums

Wang, Tingting

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 655-661.

Résumé

For the general modulo $q\ge 3$ and a general multiplicative character $\chi$ modulo $q$, the upper bound estimate of $|S(m,n,1,\chi ,q)|$ is a very complex and difficult problem. In most cases, the Weil type bound for $|S(m,n,1,\chi ,q)|$ is valid, but there are some counterexamples. Although the value distribution of $|S(m,n,1,\chi ,q)|$ is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for $k$-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet $L$-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y. Yi, X. He: On the $2k$-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.

Détail

DOI : 10.1007/s10587-012-0057-y

Classification : 11L05, 11M06, 11M38
Mots-clés : general $k$-th Kloosterman sum; Dirichlet $L$-function; the mean square value; asymptotic formula