Equivalences to the Riemann Hypothesis
Edited by Brian Conrey and David Farmer
The Riemann Hypothesis(RH) is the assertion that all of the nontrivial zeros of the Riemann zeta function have real part equal to $\frac12$.
The Riemann Hypothesis has been shown to be equivalent to an astounding variety of statements in several different areas of mathematics. Some of those equivalences are nearly trivial. For example, RH is equivalent to the nonvanishing of $\zeta(s)$ in the half-plane $\sigma>\frac12$. Other equivalences appear surprising and deep. Examples of both kinds are collected below.