On a Generalized Moment Integral containing Riemann's Zeta Function: Analysis and Experiment

TL;DR

This paper investigates a generalized moment integral related to the Riemann zeta function, combining analytical predictions with numerical experiments to reveal discontinuities, derivative properties, and potential periodicities in the integrand.

Contribution

It introduces a novel integral related to the zeta function, analytically predicts its properties, and numerically verifies these findings, including derivative behavior and periodicity indications.

Findings

Finite, discontinuous integral values predicted and verified.

Derivative of the integral relates to a generalized Dirac comb without limits.

Numerical evidence suggests the integrand exhibits quasi-periodic behavior.

Abstract

Here, we study both analytically and numerically, an integral $Z(\sigma,r)$ related to the mean value of a generalized moment of Riemann's zeta function. Analytically, we predict finite, but discontinuous values and verify the prediction numerically, employing a modified form of Ces\`aro summation. Further, it is proven and verified numerically that for certain values of $\sigma$, the derivative function $Z^{\prime}(\sigma,n)$ equates to one generalized tine of the Dirac comb function without recourse to the use of limits, test functions or distributions. A surprising outcome of the numerical study arises from the observation that the proper integral form of the derivative function is quasi-periodic, which in turn suggests a periodicity of the integrand. This possibility is also explored and it is found experimentally that zeta function values offset (shifted) over certain segments of the imaginary complex number line are moderately auto-correlated.