Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form :L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}. where \chi is a Dirichlet character and s a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1. Otherwise, the L-function is entire. Euler product Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler pro

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Mollification of the fourth moment of Dirichlet L-functions

Raphaël Pierre Zacharias

POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN2019

On the Mordell-Weil lattice of y(2) = x(3) + bx plus t(3n+1) in characteristic 3

Gauthier Leterrier

We study the elliptic curves given by y(2) = x(3) + bx + t(3n+1) over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve u(3) + bu = v(3n+1). In this way, using the Neron-Tate height on the Mordell-Weil group, we obtain lattices in dimension 2.3(n) for every n >= 1, which improve on the currently best known sphere packing densities in dimensions 162 (case n = 4) and 486 (case n = 5). For n = 3, the construction has the same packing density as the best currently known sphere packing in dimension 54, and for n = 1 it has the same density as the lattice E-6 in dimension 6.

SPRINGER INT PUBL AG2022

The fourth moment of individual Dirichlet L-functions on the critical line

Berke Topacogullari

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.

2020

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