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
AN - SCOPUS:85030158394
VL - 30
SP - R67-R80
JO - Nonlinearity
JF - Nonlinearity
SN - 0951-7715
IS - 10
ER -