By Yoichi Motohashi

This quantity provides an authoritative, updated evaluate of analytic quantity idea. It comprises striking contributions from major foreign figures during this box. middle issues mentioned comprise the idea of zeta capabilities, spectral idea of automorphic kinds, classical difficulties in additive quantity idea akin to the Goldbach conjecture, and diophantine approximations and equations. this can be a important ebook for graduates and researchers operating in quantity thought.

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J. Alg. , 4 (1995), 281-300. A. 2 Remarks on the Analytic Complexity of Zeta Functions ENRICO BOMBIERI 0. Introduction This lecture will survey some recent results obtained in collaboration with John Friedlander [2] and discuss some problems arising from our research. The Dirichlet series for the Riemann zeta-function ((s) = 2_^ — > valid for a > 1, n=l where 5 = cr + it, can be used to compute numerically ((s) for a > 1. By absolute convergence one sees that, even for x not very large, the Dirichlet polynomial ^2n

The proof is a standard application of Littlewood's lemma. r/logp. Their number is p<\ogx again by the prime number theorem. 1) is asymptotically sharp. 30 E. Bombieri Now we briefly sketch the proof of Theorem 1, referring to [2] for details. The idea is to use hypothesis (H3) to show that L(s) behaves most of the time almost as if one had a Riemann hypothesis at our disposal, save for an exceptional set of small measure. 2) Since L(s) behaves in a horizontal strip at a good interval almost as if one had a Riemann hypothesis, application of Littlewood's lemma shows that O(82))ATlogT.

Equally importantly, Selberg showed how the logarithms logL(^ + it) of 'independent' (in a sense to be clarified later on) L-functions are also statistically independent. These results have applications to the study of the distribution of zeros of certain classes of Dirichlet series, which will be examined in this paper; detailed proofs can be found in [1] and [2]1. Main result We work in the moderately general setting of the paper [1] of Bombieri and Hejhal, and consider iV functions Li(s),...