Macroscopic pair correlation of the Riemann zeroes for smooth test functions

Details Macroscopic pair correlation of the Riemann zeroes for smooth test functions Macroscopic pair correlation of the Riemann zeroes for smooth test functions

2012-03-15

arxiv.org/abs/1203.3275v2

Mathematics

Number Theory

18 pages, a typo corrected

Scientific paper

On the assumption of the Riemann hypothesis, we show that over a class of sufficiently smooth test functions, a measure conjectured by Bogolomony and Keating coincides to a very small error with the actual pair correlation measure for zeroes of the Riemann zeta function. Our result extends the well known result of Montgomery that over the same class of test functions the pair correlation measure coincides (to a larger error term) with that of the Gaussian Unitary Ensemble (GUE). The restriction of test functions remains stringent, but we are nonetheless able to detect, at a microscopically blurred resolution, macroscopic troughs in the pair correlation measure.

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