Related papers: Value Distribution of L-functions

The goal of this note is to bring attention to an interesting family of rings: the rings of $\mathbb Z$-valued functions on $\mathbb Z$ and, more generally, infinite subsets of $\mathbb Z$ whose restrictions to all finite sets are given by…

Let $f$ be a modular form which is non-ordinary at $p$. Kim and Loeffler have recently constructed two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, they showed that, as in the one-variable case, Pollack's…

We define "splitting functions of level l" for any integer l>0. These functions generalize Dwork's splitting functions : they allow us to represent additive characters of order $p^l$. Then we use these functions to obtain a Stickleberger…

We study the value-distribution of the Riemann zeta-function and related functions on and near the critical line. Amongst others, we focus on the following: The critical line is a natural boundary of the Voronin-type universality property…

Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an…

The class of Riemann zeta distribution is one of the classical classes of probability distributions on R. Multidimensional Shintani zeta function is introduced and its definable probability distributions on R^d are studied. This class…

Given a ring $R$, the notion of Sylvester rank function was conceived within the context of Cohn's classification theory of epic division $R$-rings. In this paper we study and describe the space of Sylvester rank functions on certain…

This paper first surveys the connection of integrable systems of the Painleve type to various distribution functions appearing in Wigner-Dyson random matrix theory. A short discussion is then given of the appearance of these same…

In a recent important paper, Hoffstein and Hulse generalized the notion of Rankin-Selberg convolution $L$-functions by defining shifted convolution $L$-functions. We investigate symmetrized versions of their functions. Under certain mild…

We do two things. 1. As a corollary to a stronger linearisation result (Theorem A), we prove the finite Morley rank version of the Lie-Kolchin-Malcev theorem on Lie algebras (Corollary A2). 2. We classify Lie ring actions on modules of…

In this paper, a new estimate is obtained for the multinomial Mittag-Leffler function. This function was introduced by Yuri Luchko and Rudolfo Gorenflo as the fundamental solution of the ordinary differential equation of fractional discrete…

We investigate the joint distribution of $L$-functions on the line $ \sigma= \frac12 + \frac1{G(T)}$ and $ t \in [ T, 2T]$, where $ \log \log T \leq G(T) \leq \frac{ \log T}{ ( \log \log T)^2 } $. We obtain an upper bound on the discrepancy…

In this article, we show that the Riemann hypothesis for an $L$-function $F$ belonging to the Selberg class implies that all the derivatives of $F$ can have at most finitely many zeros on the left of the critical line with imaginary part…

We survey the results and the methods in the theory of universality for various zeta and $L$-functions, obtained in these forty years after the first discovery of the universality for the Riemann zeta-function by Voronin.

These notes are from the Database of Automorphic L-functions at http://www.math.rutgers.edu/~sdmiller/l-functions . They were written up by Stephen Miller, and are based on discussions with Freydoon Shahidi, Purdue University. They are…

We deeply appreciate the papers of Ivi\'c on the links between the $2k-$th moments of the Riemann zeta function and, say, $d_k$, the $k-$divisor function. More specifically, both the one bounding the $2k-$th moment with a simple average of…

We prove under GRH that zeros of $L$-functions of modular forms of level $N$ and weight $k$ become uniformly distributed on the critical line when $N+k\to\infty.$

The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to…

In a paper published in 1959, Shimura presented an elegant calculation of the critical values of L-functions attached to elliptic modular forms using the first cohomology group. We will show that a similar calculation is possible for…

Hardy's $Z$-function $Z(t)$ is a real-valued function of the real valuable $t$, and its zeros exactly correspond to those of the Riemann zeta-function on the critical line. In 2012, K.~Matsuoka showed that for any non-negative integer $k$,…