The Integral of the Riemann xi-function

TL;DR

This paper investigates the zeros of the integral of the Riemann xi-function and a related family of Fourier integral functions, revealing finite zeros on the critical line and their unbounded deviation from it, along with an infinite analogue of the de Bruijn-Newman constant.

Contribution

It introduces a new family of functions satisfying a functional equation, analyzes their zeros, and defines an infinite de Bruijn-Newman constant for this family.

Findings

The integral of the xi-function has exactly one zero on the critical line.

Members of the family have finitely many zeros on the critical line.

Zeros can be arbitrarily far from the critical line.

Abstract

This paper studies the integral of the Riemann xi-function. More generally, it studies a one-parameter family of functions given by Fourier integrals and satisfying a functional equation. Members of this family are shown to have only finitely many zeros on the critical line, with the integral of the Riemann xi-function having exactly one zero on the critical line, at s = 1/2. The zeros of the integral of the xi-function are shown to lie arbitrarily far away from the critical line. An analogue of the de Bruijn-Newman constant is introduced for this family, and shown to be infinite.