• Yulong Zhang

      Articles written in Proceedings – Mathematical Sciences

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

      Rong Ma Junhuai Zhang Yulong Zhang

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      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.

    • A Note on a Kind of Character Sums over the Short Interval

      Rong Ma Yulong Zhang

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      Let 𝑝 be a prime, 𝜒 denote the Dirichlet character modulo 𝑝 and $L(p)=\{a\in\mathbb{Z}^+|(a,p)=1,aā\equiv 1(\mathrm{mod} p),|a - ā|\leq H\}$. We study the distribution of elements in the set $L(p)$ in character over the short interval. In this paper, we use the analytic method and show the distribution property of

      \begin{equation*}\sum_{\substack{n\leq N}\\{n\in L(p)}}\chi(n),\end{equation*}

      and give a non-trivial estimate.

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