The Integral of the Riemann xi-function

Details The Integral of the Riemann xi-function The Integral of the Riemann xi-function

2011-06-21

arxiv.org/abs/1106.4348v3

Commentarii Mathematici Universitatis Sancti Pauli 60 (2011), No. 1-2, 143--169

Mathematics

Number Theory

23 pages, LaTeX; v2 26 pages, substantive changes to details

Scientific paper

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.

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