Related papers: Joint universality for dependent $L$-functions

Given a Dirichlet character $\chi$ modulo $q$ and its associated $L$-function, $L(s,\chi)$, we provide an explicit version of Burgess' estimate for $|L(s, \chi)|$. We use partial summation to provide bounds along the vertical lines $\Re{s}…

For $0<\alpha, \lambda \leq 1$, the Lerch zeta-function is defined by $L(s;\alpha, \lambda)$$:= \sum_{n=0}^\infty e^{2\pi i\lambda n} (n+\alpha)^{-s}$, where $\sigma>1$. In this paper, we prove joint universality for Lerch zeta-functions…

We propose a numerical method for approximating and discovering zeros of the Dirichlet L-function L(s, chi) corresponding to real Dirichlet characters chi.

In the paper we introduce the new approach how to use an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.

We consider a general form of L-function L(s) defined by an Euler product and satisfies several analytic assumptions. We show several asymptotic formulas for L(1) and log L(1). In particular those asymptotic formulas are valid for Dirichlet…

In this paper, we investigate the conditional large values of the quadratic Dirichlet $L$-functions near the central point $s=1/2$. When $\sigma $ closes to $1/2$ within a suitable range, we show that $L(\sigma, \chi_d)$ have the…

We study the relation between the size of $L(1,\chi)$ and the width of the zero-free interval to the left of that point.

Recently, the two variable $q$-$L$-functions which interpolate the generalized $q$-Bernoulli polynomials associated with $\chi$ are introduced and studied, cf. [2]. In this paper, we construct multiple Dirichlet's $q$-$L$-function which…

We establish sharp upper bounds for shifted moments of quadratic Dirichlet $L$-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.

In the present work we show that the Dirichlet series with the Euler product having analytical continuation to the critical strip without singularities, in some natural conditions, can be approximated by partial products of Euler type in…

We give a combinatorial proof of the transcendence of $L(1,\chi_s)/\Pi$, where $L(1,\chi_s)$ (resp. $\Pi$) is the analogue in characteristic $p$ of the function $L$ of Dirichlet (resp. $\pi$). This result has been proven by G. Damamme using…

First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of $L$-functions from the Selberg class $\mathcal{S}$. We also study the latter abscissa inside the extended Selberg class,…

Let $A$ be a rational function. For any decomposition of $A$ into a composition of rational functions $A=U\circ V$ the rational function $\widetilde A=V\circ U$ is called an elementary transformation of $A$, and rational functions $A$ and…

The Euler product formula relates Dirichlet $L(s,\chi)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re…

We establish upper bounds for shifted moments of cubic and quartic Dirichlet $L$-functions under the generalized Riemann hypothesis. As an application, we prove bounds for moments of cubic and quartic Dirichlet character sums.

Let $\chi$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s, \chi)$ denote its associated Dirichlet $L$-function. We show that $|L(1, \chi)| \geq 1/(9.12255 \log(q/\pi))$ for sufficiently large $q$, and…

Let $\chi$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log |L(s, \chi)|$ in $q$-aspect at a fixed $s=\sigma>1/2$, under Generalized Riemann Hypothesis (GRH), we explain…

In a recent work arXiv:2004.14450, it has been shown that $L$-functions associated with arbitrary non-zero cusp forms take large values at the central critical point. The goal of this note is to derive analogous results for twists of…

We look at the values of two Dirichlet $L$-functions at the Riemann zeros (or a horizontal shift of them). Off the critical line we show that for a positive proportion of these points the pairs of values of the two $L$-functions are…

This text shows the existence of large (3.54 times the average) gaps between consecutive zeros, on the critical line, of some Dirichlet $L$-functions $L(s,\chi),$ with $\chi$ being an even primitive Dirichlet character.