Scalar modular bootstrap and zeros of the Riemann zeta function

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.

Automated summary

By utilizing harmonic analysis, this article derives a crossing equation that applies only to scalar primary operators in any two-dimensional conformal field theory with U(1)(c) symmetry. The obtained crossing equation provides bounds on the scalar gap of all such theories and remarkably includes information about nontrivial zeros of the Riemann zeta function. Consequently, the Riemann hypothesis is rephrased solely as a statement regarding the asymptotic density of scalar operators in certain two-dimensional conformal field theories. Generalizations to theories with only Virasoro symmetry are also discussed.