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

TL;DR

This paper links the ratios conjecture for the Riemann zeta-function with conjectures on divisor sum correlations, showing they imply each other and thus support the ratios conjecture with new theoretical backing.

Contribution

It establishes a connection between the ratios conjecture and divisor sum correlations, generalizing previous specific cases and providing new support for the conjecture.

Findings

Ratios conjecture and divisor sum correlation conjecture imply each other.

Generalization of a recent two-over-two ratios calculation.

Provides new theoretical evidence for the ratios conjecture.

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.