Theta and Riemann xi function representations from harmonic oscillator eigensolutions

TL;DR

This paper derives new integral representations of the Riemann xi function using harmonic oscillator eigensolutions, extending the functional equation of the Riemann zeta function and generalizing the Riemann-Siegel formula.

Contribution

It introduces a novel approach connecting quantum harmonic oscillator solutions to the properties of the Riemann zeta and xi functions.

Findings

Extended functional equation for Riemann zeta

New integral representations of Riemann xi function

Generalization of the Riemann-Siegel integral formula

Abstract

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical zeta function. A key result provides a basis for generalizing the important Riemann-Siegel integral formula.