String theory, N=4 SYM and Riemann hypothesis

We discuss new relations among string theory, four-dimensional N=4 supersymmetric Yang-Mills theory (SYM) and the Riemann hypothesis.
It is known that the Riemann hypothesis is equivalent to an inequality for the sum of divisors function σ(n).
Based on previous results in literature, we focus on the fact that σ(n) appears in a problem of counting supersymmetric states in the N=4 SYM with SU(3) gauge group:
the Schur limit of the superconformal index plays a role of a generating function of σ(n). Then assuming the Riemann hypothesis gives bounds on information on the 1/8-BPS states in the N=4 SYM. The AdS/CFT correspondence further connects the Riemann hypothesis to the type IIB superstring theory on AdS5×S5.
In particular, the Riemann hypothesis implies a miraculous cancellation among Kaluza-Klein modes of the supergravity multiplet and D3-branes wrapping supersymmetric cycles in the string theory.
We also discuss possibilities to gain new insights on the Riemann hypothesis from the physics side.
This talk is based on a collaboration with Takuya Yoda (arXiv:220317091).