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A pair correlation hypothesis and the exceptional set in Goldbach's problem

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

A. Languasco  and
A. Perelli
A. Languasco
Affiliation:
Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy. e-mail: [email protected]
A. Perelli
Affiliation:
Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy, e-mail: [email protected]
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Extract

In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.

MSC classification

Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes
Type
Research Article
Information
Mathematika , Volume 43 , Issue 2 , December 1996 , pp. 349 - 361
Copyright
Copyright © University College London 1996

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