Self-intersections of the Riemann zeta function on the critical line

TL;DR

This paper proves that the Riemann zeta function has only countably many self-intersections on the critical line, and extends the result to more general analytic functions that are locally injective on the line.

Contribution

It establishes a countability result for self-intersections of the Riemann zeta function and generalizes to analytic functions with local injectivity near the critical line.

Findings

Riemann zeta function has only countably many self-intersections on the critical line.

For analytic functions locally injective on the line, self-intersections are either countable or the image is a loop.

The result applies broadly to functions with similar properties near the critical line.

Abstract

We show that the Riemann zeta function \zeta\ has only countably many self-intersections on the critical line, i.e., for all but countably many z in C the equation \zeta(1/2+it)=z has at most one solution t in R. More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on R, then either the set {(a,b) in R^2:a \ne b and F(a)=F(b)} is countable, or the image F(R) is a loop in C.