An Integral Equation for Riemann's Zeta Function and its Approximate Solution

Abstract

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann's $\xi (s)$ function, and thus $\zeta (s)$ indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates $\zeta (s)$ anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, I obtain both an analytic expression for $\zeta (\sigma +it)$ everywhere inside the asymptotic ($t\to \infty )$ critical strip, and an approximate solution, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of $\zeta (\sigma +it)$ for different values of $\sigma$ and equal values of $t$; this is illustrated in a number of Figures.

This version is extensively revised, reorganized, modified and corrected. 37 pages, 17 Figures

References in corpus (3)

• Integral and Series Representations of Riemann's Zeta function, Dirichelet's Eta Function and a Medley of Related Results
• An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N
• Exploring Riemann's Functional Equation

Cited by in corpus (1)

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