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

Brian Conrey, Jonathan P. Keating

Type: Article

Publication Date: 2017-09-15

Citations: 4

DOI: https://doi.org/10.1088/1361-6544/aa8423

Abstract

We establish a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of M\"{o}bius 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.

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arXiv (Cornell University) - View - PDF
Bristol Research (University of Bristol) - View - PDF
Oxford University Research Archive (ORA) (University of Oxford) - View - PDF
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+	L-functions and the characteristic polynomials of random matrices	2005	JP Keating
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+	A quadratic divisor problem	1994	William Duke John Friedlander Henryk Iwaniec
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+	Gutzwiller's Trace Formula and Spectral Statistics: Beyond the Diagonal Approximation	1996	E. Bogomolny J. P. Keating