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

    • Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      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

       

      Rong Ma1 Junhuai Zhang1 Yulong Zhang2

      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
    • Dates

       
  • Proceedings – Mathematical Sciences | News

© 2017-2019 Indian Academy of Sciences, Bengaluru.