An extremal problem related to generalizations of the Nyman-Beurling and B\'aez-Duarte criteria

TL;DR

This paper generalizes criteria related to the absence of zeros of Dirichlet L-functions in certain half-planes, by solving an extremal problem involving Dirichlet polynomials that take specific values at these zeros.

Contribution

It introduces new generalizations of the Nyman-Beurling and Be1ez-Duarte criteria and solves a natural extremal problem for Dirichlet polynomials associated with L-function zeros.

Findings

Generalized criteria for zeros of Dirichlet L-functions in b1 > 1/p

Solved an extremal problem for Dirichlet polynomials at zeros

Established conditions linking polynomial behavior to L-function zeros

Abstract

We establish generalizations of the Nyman-Beurling and B\'aez-Duarte criteria concerning lack of zeros of Dirichlet $L$-functions in the semi-plane $\Re(s) >1/p$ for $p\in (1,2]$. We pose and solve a natural extremal problem for Dirichlet polynomials which take values one at the zeros of the corresponding $L$-function on the vertical line $\Re(s)=1/p$.