A proof of the Riemann hypothesis

TL;DR

This paper proves the Riemann hypothesis by demonstrating the nonnegativity of a specific operator's trace, which implies all nontrivial zeros of the zeta function lie on the critical line, confirming the hypothesis.

Contribution

It provides a novel proof of the Riemann hypothesis by analyzing traces of an integral operator on $L^2$ space and establishing their nonnegativity.

Findings

Trace of the operator on one subspace is zero.

Trace of the operator on the other subspace is nonnegative.

Nonnegative trace implies the Riemann hypothesis via Li's criterion.

Abstract

In this paper we study traces of an integral operator on two orthogonal subspaces of a $L^2$ space. One of the two traces is shown to be zero. Also, we prove that the trace of the operator on the second subspace is nonnegative. Hence, the operator has a nonnegative trace on the $L^2$ space. This implies the positivity of Li's criterion. By Li's criterion, all nontrivial zeros of the Riemann zeta-function lie on the critical line.