The Riemann hypothesis for certain integrals of Eisenstein series

Details The Riemann hypothesis for certain integrals of Eisenstein series The Riemann hypothesis for certain integrals of Eisenstein series

2004-12-02

arxiv.org/abs/math/0412039v4

J. Number Theory 118 (2006) 98-122

Mathematics

Number Theory

23 pages; corrected residues of functions in theorem 1 and 2, added reference; v3 small corrections, removed uncited reference

Scientific paper

This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical line Re(s) = 1/2. For the constant term of the Eisenstein series it shows that all zeros are on the critical line for fixed y= Im(z) \ge 1, except possibly for two real zeros, which are present if and only if y > 4 \pi e^{-\gamma} = 7.0555+. It shows the Riemann hypothesis holds for all truncation integrals with truncation parameter T \ge 1. For T=1 this proves the Riemann hypothesis for a zeta function recently introduced by Lin Weng, attached to rank 2 semistable lattices over the rationals.

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