Averages of ratios of the Riemann zeta-function and correlations of divisor sums

Nonlinearity has published articles containing a significant number-theoretic component since the journal was first established. We examine one thread, concerning the statistics of the zeros of the Riemann zeta function. We extend this by establishing a connection between the ratios conjecture for t...

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Bibliographic Details
Published in:Nonlinearity 2017-10, Vol.30 (10), p.R67-R80
Main Authors: Conrey, Brian, Keating, Jonathan P
Format: Article
Language:English
Subjects:
correlations of divisor sums
Riemann zeta-function
significant number-theoretic component
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Summary:Nonlinearity has published articles containing a significant number-theoretic component since the journal was first established. We examine one thread, concerning the statistics of the zeros of the Riemann zeta function. We extend this by establishing a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/aa8423