Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function

TL;DR

This paper connects the zeros of the Riemann zeta function to bounds on scalar operators in 2D conformal field theories with $U(1)^c$ symmetry, offering a novel approach to the Riemann hypothesis.

Contribution

It introduces a crossing equation derived via harmonic analysis that encodes information about the Riemann zeta zeros within conformal field theory constraints.

Findings

Derived bounds on scalar gaps in 2D CFTs with $U(1)^c$ symmetry.

Rephrased the Riemann hypothesis as a statement about scalar operator density.

Established a link between zeta zeros and conformal bootstrap equations.

Abstract

Using the technology of harmonic analysis, we derive a crossing equation that acts only on the scalar primary operators of any two-dimensional conformal field theory with $U(1)^c$ symmetry. From this crossing equation, we derive bounds on the scalar gap of all such theories. Rather remarkably, our crossing equation contains information about all nontrivial zeros of the Riemann zeta function. As a result, we rephrase the Riemann hypothesis purely as a statement about the asymptotic density of scalar operators in certain two-dimensional conformal field theories. We discuss generalizations to theories with only Virasoro symmetry.