The Riemann hypothesis for certain integrals of Eisenstein series

This paper studies the nonholomorphic Eisenstein series E(z,s) for the modular surface PSL(2,Z)\H, and shows that integration with respect to certain nonnegative measures μ(z) gives meromorphic functions Fμ(s) that have all their zeros on the line . 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−γ=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.