• On the $2m$-th Power Mean of Dirichlet 𝐿-Functions with the Weight of Trigonometric Sums

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/119/04/0411-0421

• # Keywords

Dirichlet 𝐿-functions; trigonometric sums; congruence equation; asymptotic formula.

• # Abstract

Let 𝑝 be a prime, 𝜒 denote the Dirichlet character modulo $p,f(x)=a_0+a_1 x+\cdots+a_kx^k$ is a 𝑘-degree polynomial with integral coefficients such that $(p, a_0,a_1,\ldots,a_k)=1$, for any integer 𝑚, we study the asymptotic property of

\begin{equation*}\sum\limits_{\chi\neq \chi_0}\left| \sum\limits^{p-1}_{a=1}\chi(a)e\left( \frac{f(a)}{p}\right)\right|^2 |L(1,\chi)|^{2m},\end{equation*}

where $e(y)=e^{2\pi iy}$. The main purpose is to use the analytic method to study the $2m$-th power mean of Dirichlet 𝐿-functions with the weight of the general trigonometric sums and give an interesting asymptotic formula. This result is an extension of the previous results.

• # Author Affiliations

1. School of Science, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China
2. The School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019