Related papers: A Note on the Mean Value of $L$--functions in Func…

Let $\chi$ be a Dirichlet character modulo $q$, let $L(s, \chi)$ be the attached Dirichlet $L$-function, and let $L^\prime(s, \chi)$ denotes its derivative with respect to the complex variable $s$. The main purpose of this paper is to give…

In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields $\mathbb{F}_q$, and study the resulting AGM graph with points $(a,b) \in \mathbb{F}_q \times \mathbb{F}_q$ and directed edges…

We prove some general asymptotic formula for the values of $L$-function of a sequence of constructible $\mathbb Q_l$-sheaves on curves over $\mathbb F_q$ with some good asymptotic properties. We also give the asymptotic formula for the…

In this paper, we study the value distribution of the derivative of a Dirichlet $L$-function $L'(s,\chi)$ at the $a$-points $\rho_{a,\chi}=\beta_{a,\chi}+i\gamma_{a,\chi}$ of $L(s,\chi).$ We give an asymptotic formula for the sum…

The first and second moments are established for the family of quadratic Dirichlet $L$--functions over the rational function field at the central point $s=\tfrac{1}{2}$ where the character $\chi$ is defined by the Legendre symbol for…

We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.

Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $\chi$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,\chi)$ and $L(4,\chi)$ using certain…

This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.

For Dirichlet characters $\chi$ mod $k$ where $k\geq 3$, we here give a computable formula for evaluating the mean square sums $\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for any positive integer $r\geq 3$.…

Let $\chi$ be a quadratic Dirichlet character. In some literatures, various asymptotic formulae of $L'(1,\chi)$, under the assumption that $L(1,\chi)$ takes a small value, were derived. In this paper, we will give a new treatment unified…

We prove an asymptotic formula for the mean-square average of $L$- functions associated to subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\cal A}(p,d)$ recently introduced by E.…

Let $m\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\frac{2}{\phi(f)}\sum_{\chi (-1)=(-1)^m}\vert L(m,\chi )\vert^2,$$ where $\chi$ ranges over the $\phi (f)/2$ Dirichlet characters modulo $f>2$ with the…

Let $\gcd(k,j)$ be the greatest common divisor of the integers $k$ and $j$. For any arithmetical function $f$, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is…

In this paper we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of $L$-functions. We also adapt to the function setting the heuristics first…

We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is…

We establish asymptotic estimates for exact upper bounds of uniform approximations by Fourier sums on the classes of $2\pi$-periodic functions, which are represented by convolutions of functions $\varphi (\varphi\bot 1)$ from unit ball of…

We obtain asymptotic formulas for the second and third moment of quadratic Dirichlet $L$--functions at the critical point, in the function field setting. We fix the ground field $\mathbb{F}_q$, and assume for simplicity that $q$ is a prime…

Let $f$ be a holomorphic cusp form of weight $k$ with respect to full modular group $SL_2(\mathbb{Z})$ satisfying a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. Good gave the approximate functional…

We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2\pi$-periodic functions, representable in the form of convolutions of functions $\varphi$, which belong to unit balls…

Let $q\ge 2$ and $N\ge 1$ be integers. W. Zhang (2008) has shown that for any fixed $\epsilon> 0$, and $q^{\epsilon} \le N \le q^{1/2 -\epsilon}$, $$ \sum_{\chi \ne \chi_0} |\sum_{n=1}^N \chi(n)|^2 |L(1, \chi)|^2 = (1 + o(1)) \alpha_q q N…