Properties of scalar partition functions of 2d CFTs

We study the spectrum of scalar primary operators in any two-dimensional conformal field theory. We show that the scalars alone obey a nontrivial crossing equation. This extends previous work that derived a similar equation for Narain conformal field theories. Additionally, we show that at high temperature, the difference between the true scalar partition function and the one predicted from a semiclassical gravity calculation is controlled by: the modular integral of the partition function, the light states of the theory, and an infinite series terms directly related to the nontrivial zeros of the Riemann zeta function. We give several numerical examples and compute their modular integrals.