Skip to content

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

Research output: Contribution to journalArticle

Standard

Averages of ratios of the Riemann zeta-function and correlations of divisor sums. / Conrey, Brian; Keating, Jonathan P.

In: Nonlinearity, Vol. 30, No. 10, 15.09.2017, p. R67-R80.

Research output: Contribution to journalArticle

Harvard

APA

Vancouver

Author

Conrey, Brian ; Keating, Jonathan P. / Averages of ratios of the Riemann zeta-function and correlations of divisor sums. In: Nonlinearity. 2017 ; Vol. 30, No. 10. pp. R67-R80.

Bibtex

@article{a80143b0f7354c0e9f7e47748bedb966,
title = "Averages of ratios of the Riemann zeta-function and correlations of divisor sums",
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 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.",
keywords = "correlations of divisor sums, Riemann zeta-function, significant number-theoretic component",
author = "Brian Conrey and Keating, {Jonathan P.}",
year = "2017",
month = "9",
day = "15",
doi = "10.1088/1361-6544/aa8423",
language = "English",
volume = "30",
pages = "R67--R80",
journal = "Nonlinearity",
issn = "0951-7715",
publisher = "IOP Publishing",
number = "10",

}

RIS - suitable for import to EndNote

TY - JOUR

T1 - Averages of ratios of the Riemann zeta-function and correlations of divisor sums

AU - Conrey, Brian

AU - Keating, Jonathan P.

PY - 2017/9/15

Y1 - 2017/9/15

N2 - 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.

AB - 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.

KW - correlations of divisor sums

KW - Riemann zeta-function

KW - significant number-theoretic component

UR - http://www.scopus.com/inward/record.url?scp=85030158394&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/aa8423

DO - 10.1088/1361-6544/aa8423

M3 - Article

VL - 30

SP - R67-R80

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 10

ER -