相关论文: Non-vanishing of Dirichlet series with periodic co…

We address the question of non-vanishing of $L(1,f)$ where $f$ is an algebraic-valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions $f$ for which $L(1,f)=0$. The case of odd functions was…

Given $k\in\mathbb N$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\sum_{n\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a periodic function modulo a prime $p$. We show that if $(k,p-1)=1$ or $(k,p-1)=2$ and…

In a written correspondence with A. Livingston, Erd\H{o}s conjectured that for any arithmetical function $f$, periodic with period $q$, taking values in $\{-1,1\}$ when $q \nmid n$ and $f(n)=0$ when $q \mid n$, the series…

In this paper, we give Dirichlet series with periodic coefficients that have Riemann's functional equation and real zeros of Dirichlet $L$-functions. The details are as follows. Let $L(s,\chi)$ be the Dirichlet $L$-function and $G(\chi)$ be…

The class of Dirichlet series associated with a periodic arithmetical function $f$ includes the Riemann zeta-function as well as Dirichlet $L$-functions to residue class characters. We study the value-distribution of these Dirichlet series…

We prove that given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $\chi$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes \chi)$…

Let $a=(a_n)_{n\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\sum_{n\ge 1} a_n/n^s$, and $N_a(\sigma_1, \sigma_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $\sigma_1…

We present a simple analytic proof that L-functions of real non-principal Dirichlet characters are nonzero at 1.

We prove that an ordinary Dirichlet series with coefficients a(n)=g(n b) has an abscissa of convergence 0 if g is an odd 1-periodic, real-analytic function and b is Diophantine. We also show that if g is odd and has bounded variation and b…

We investigate the consequences of natural conjectures of Montgomery type on the non-vanishing of Dirichlet $L$-functions at the central point. We first justify these conjectures using probabilistic arguments. We then show using a result of…

In this paper we consider certain 1-parametric family of Dirichlet series. For a particular value of the parameter the series turns into the Dirichlet series for the Riemann zeta function. We prove that almost every series of the family has…

In this paper, we prove the simultaneous non-vanishing of four Dirichlet $L$-functions at any point on the critical line. More precisely, let $\chi_1,\ldots,\chi_4$ be even Dirichlet characters modulo $D_1,\ldots, D_4$ respectively, where…

We study multiplicative functions $f$ satisfying $|f(n)|\le 1$ for all $n$, the associated Dirichlet series $F(s):=\sum_{n=1}^{\infty} f(n) n^{-s}$, and the summatory function $S_f(x):=\sum_{n\le x} f(n)$. Up to a possible trivial…

Fix $g$ a Hecke-Maass form for $SL_3(\mathbb{Z})$. Let $q$ be any large prime number. In the family of holomorphic newforms $f$ of level $q$ and fixed weight, we find the average value of the product $L(\half,g\times f)L(\half,f)$. From…

Given a large, square-free, smooth conductor, we establish the non-vanishing of the central values for at least $35.9\%$ of the primitive Dirichlet $L$-functions.

We prove that for at least $\frac{7}{19}$ of the primitive Dirichlet characters $\chi$ with large general modulus, the central value $L(\frac12,\chi)$ is non-vanishing.

Let $\chi$ be a primitive Dirichlet character modulo $q$ and $L(s,\chi)$ be the Dirichlet L-function associated to $\chi$. Using a new two-piece mollifier we show that $L(\tfrac{1}{2},\chi)\ne0$ for at least 34% of the characters in the…

We study non-vanishing of Dirichlet $L$-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if $\psi$ is a real primitive character modulo $D \in…

We establish an asymptotic formula for the first moment and derive an upper bound for the second moment of L-functions associated with the complete family of primitive cubic Dirichlet characters defined over the Eisenstein field. Our…

In an attempt to resolve a folklore conjecture of Erd\H{o}s regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{-1, 1 \}$, Livingston conjectured the…