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Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function
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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.
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Properties of scalar partition functions of 2d CFTs
Scalar Virasoro primaries in any 2d CFT obey a crossing equation whose high-temperature form is controlled by a modular integral and by oscillations tied to the nontrivial zeros of the Riemann zeta function.
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