On a Bessel function series related to the Riemann xi function

TL;DR

This paper derives new Fourier-type integral identities for the Riemann xi function $\Xi(y)$ that are tied to a Bessel function series $J_0(rny)$ and a confluent hypergeometric term ${}_1F_1\left(\cdot;1;-\frac{r^2}{4t}\right)$. The principal result expresses an integral involving $\Xi(y)$ and $F^{+}\left(x,\tfrac{1}{2}+iy\right)$ as a Bessel-sum $\sum_{n\ge1} e^{-t n^2 e^{-2x}} J_0\left(r n e^{-x}\right)$ minus a ${}_1F_1$ correction, derived via Mellin-transform methods and the xi-functional equation. It also provides an integral representation for $H_1(y)$ that generalizes the Jacobi theta functional equation (reducing to the classical case when $r=0$). Moreover, the same Bessel-series identity yields a solution to a cylindrical heat-equation boundary-value problem with boundary data reflecting theta-function relations, highlighting a link between analytic number theory and PDE boundary-value analysis. The techniques combine Parseval's theorem for Mellin transforms, contour integration, and the Riemann xi-function's functional equation to obtain the main results.

Abstract

We establish new Fourier integral evaluations involving the Riemann xi function related to a series involving Bessel function of the first kind. We show this infinite series involving the Bessel function of the first kind solves a boundary value problem for the cylindrical heat equation.

Theorems & Definitions (7)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 3.1
• proof : Proof of Theorem 1.1
• proof : Proof of Theorem 1.2
• proof : Proof of Theorem 1.3