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

Michael Milgram

Published 2019-01-06Version 1

Two obscure 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 $\zeta(\sigma+i\rho)$ everywhere inside the asymptotic ($\rho\rightarrow\infty)$ 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 $\zeta(\sigma+i\rho)$ for different values of $\sigma$ and equal values of $\rho$; this is illustrated in a number of Figures. Finally, arguments are presented to show that that the proffered "approximate" solution is less approximate and more rigorous than meets the eye.