The Integral of the Riemann xi-function

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.