The Jensen–Pólya program for various L-functions

Abstract

Abstract Pólya proved in 1927 that the Riemann hypothesis is equivalent to the hyperbolicity of all of the Jensen polynomials of degree d and shift n for the Riemann Xi-function. Recently, Griffin, Ono, Rolen, and Zagier [M. Griffin, K. Ono, L. Rolen and D. Zagier, Jensen polynomials for the Riemann zeta function and other sequences, Proc. Natl. Acad. Sci. USA 116 2019, 23, 11103–11110] proved that for each degree d ≥ 1 {d\geq 1} all of the Jensen polynomials for the Riemann Xi-function are hyperbolic except for possibly finitely many n. Here we extend their work by showing that the same statement is true for suitable L-functions. This offers evidence for the generalized Riemann hypothesis.