The Riemann hypothesis for certain integrals of Eisenstein series

✍ Scribed by Jeffrey C. Lagarias; Masatoshi Suzuki

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 204 KB
• Volume: 118
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2005.08.010

✦ Synopsis

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 F μ (s) that have all their zeros on the line (s) = 1 2 . For the constant term a 0 (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 Z 2,Q (s) recently introduced by Lin Weng, associated to rank 2 semistable lattices over Q.