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Two identities extracted from the literature are coupled to obtain anintegral equation for Riemann's $\xi(s)$ function, and thus $\zeta(s)$indirectly. The equation has a number of simple properties from which usefulderivations flow, the most notable of which relates $\zeta(s)$ anywhere in thecritical strip to its values on a line anywhere else in the complex plane. Fromthis, I obtain both an analytic expression for $\zeta(\sigma+it)$ everywhereinside the asymptotic ($t\rightarrow\infty)$ critical strip, and an approximatesolution, within the confines of which the Riemann Hypothesis is shown to betrue. The approximate solution predicts a simple, but strong correlationbetween the real and imaginary components of $\zeta(\sigma+it)$ for differentvalues of $\sigma$ and equal values of $t$; this is illustrated in a number ofFigures.

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