On Repeated Values of the Riemann Zeta Function on the Critical Line

Abstract

Let $\zeta (s)$ be the Riemann zeta function. In this paper, we study repeated values of $\zeta (s)$ on the critical line, and we give evidence to support our conjecture that for every nonzero complex number $z$, the equation $\zeta (1/2 + i t) = z$ has at most two solutions $t \in R$. We prove a number of related results, some of which are unconditional, and some of which depend on the truth of the Riemann hypothesis. We also propose some related conjectures that are implied by Montgomery’s pair correlation conjecture.