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, both an analytic expression for $\zeta (\sigma +it)$, everywhere inside the asymptotic $(t\to \infty )$ 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 $\zeta (\sigma +it)$ for different values of $\sigma$ and equal values of $t$; this is illustrated in a number of figures.