Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function

We derive a crossing equation that acts only on the scalar primary operators of any 2d CFT with U(1)^c symmetry. 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 as a statement about the asymptotic density of scalar operators in certain theories. We discuss generalizations to theories with only Virasoro symmetry. Based on 2208.02259 with C.-H. Chang.