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The number of zeros in a horizontal strip
Some mean value theorems
2 other sections not shown
absolute convergence analytic continuation apply lemma arbitrary automorphic form chapter character mod q class of functions conditions of theorem converges absolutely converges somewhere critical line cusp form defined denote the number Dirichlet L-functions Dirichlet series Dirichlet series Xa Epstein zeta-functions estimate exists a positive finite order form of dimension form with signature Fourier function p(s functional equation Further given signature Hecke Hence p(s holds identically zero interval Let g(z Let h Let p(s mean value modular form modular group No(T non-real zeros number of zeros O(log Phragmén-Lindelöf theorem positive constant positive integer positive number possesses property principal value proof of theorem real coefficients real number relation represent an integral result Riemann hypothesis right half-plane satisfies series for p(s series oc ſº strip Suppose that p(s theorem 13 Tº T(s transformation valid Y)-function p(s zeros of p(s