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 ξs function and thus ζs indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζs anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for ζσ+it, everywhere inside the asymptotic t⟶∞ critical strip, as well as an approximate solution can be obtained, 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 ζσ+it for different values of σ and equal values of t; this is illustrated in a number of figures.

A Note on the Riemann $ξ$-Function

“…Eq. ( 10) is equivalent to the very important eq(3.10) in Milgram's paper [5] and (13), apart from having summed a series, is LeClair's key formula (15) in [6] To explore further consequences of (2), note that the Mellin transform…”

“…This work was inspired by reading [5] and I thank its author for this opportunity. I also thank Dr. Michael Milgram and Prof. Richard Brent for valuable comments and suggestions.…”

A Series Representation for Riemann's Zeta Function and some Interesting Identities that Follow

“…In a recent paper [1], I have developed an integral equation for Riemann's function ξ(s), based on LeClair's series [2] involving the Generalized Exponential Integral. Glasser, in a subsequent paper [3], has shown that this integral equation is equivalent to the standard statement of Cauchy's Integral Theorem applied to ξ(s), from which he closed the circle [3,Eq.…”