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Keywords:
general $k$-th Kloosterman sum; Dirichlet $L$-function; the mean square value; asymptotic formula
general $k$-th Kloosterman sum; Dirichlet $L$-function; the mean square value; asymptotic formula
Summary:
For the general modulo $q\geq 3$ 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.
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