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

## TL;DR

This paper derives an integral equation for Riemann's zeta function, providing an analytic expression and an approximate solution that supports the Riemann Hypothesis, revealing correlations between the real and imaginary parts of the zeta function.

## Contribution

It introduces a novel integral equation for (s) and (s) (s) (s) (s), offering new insights and an approximate solution that bolsters the Riemann Hypothesis.

## Key findings

- Derived an integral equation relating (s) to its values elsewhere
- Provided an analytic expression for (+it) inside the critical strip
- Predicted strong correlation between real and imaginary parts of (+it) for different 

## 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 ($t\rightarrow\infty)$ 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 $\zeta(\sigma+it)$ for different values of $\sigma$ and equal values of $t$; this is illustrated in a number of Figures.