The Riemann Hypothesis for Certain Integrals of Eisenstein Series

This paper studies the non-holomorphic Eisenstein series E(z, s) for the modular surface PSL(2,Z)\H, and shows that integration with respect to certain non-negative measures μ(z) gives meromorphic functions Fμ(s) that have all their zeros on the line R(s) = 1 2 . For the constant term a0(y, s) of the Eisenstein series the Riemann hypothesis holds for all values y ≥ 1, with at most two exceptional real zeros, which occur exactly for those y > 4πe = 7.0555+. The Riemann hypothesis holds for all truncation integrals with truncation parameter T ≥ 1. At the value T = 1 this proves the Riemann hypothesis for a zeta function Z2,Q(s) recently introduced by Lin Weng, associated to rank 2 semistable lattices over Q.