The Riemann hypothesis for certain integrals of Eisenstein series (解析的整数論とその周辺 研究集会報告集)

By Jeffrey C. Lagarias, Masatoshi Suzuki. 2004

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Abstract 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 R ( 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 > 4 π e − 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 .

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/17f7b293cf35aa6803a799fcdd489a6dc786674c