Averages of ratios of the Riemann zeta-function and correlations of divisor sums. (15th September 2017)

• Record Type: Journal Article
• Authors: Conrey, Brian; Keating, Jonathan P
• Is Part Of: Nonlinearity. Volume 30:Number 10(2017:Oct.)
• Page Start: R67
• Page End: R80
• Publication Date: 2017-09-15
• Subjects: Riemann zeta-function -- correlations of divisor sums -- significant number-theoretic component; 11M50 -- 11M26 -- 11N37 -- 81Q50; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa8423
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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.