The sixth moment of the Riemann zeta function and ternary additive divisor sums

TL;DR

This paper investigates the sixth moment of the Riemann zeta function on the critical line, linking it to ternary additive divisor sums and providing new asymptotic formulas with implications for longstanding conjectures.

Contribution

It establishes an asymptotic formula for the sixth moment assuming a conjectural formula for ternary additive divisor sums, advancing understanding of zeta moments.

Findings

Proves a conjectural formula implies an asymptotic for the sixth moment.

Provides evidence for a conjecture on shifted moments of the zeta function.

Improves on previous upper bounds for the sixth moment.

Abstract

Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of the Riemann zeta function on the critical line. This provides a rigorous proof for a heuristic argument of Conrey and Gonek. Furthermore, this gives some evidence towards a conjecture of Conrey, Keating, Farmer, Rubinstein, and Snaith on shifted moments of the Riemann zeta function. In addition, this improves on a theorem of Ivic, who obtained an upper bound for the the sixth moment of the zeta function, based on the assumption of a conjectural formula for correlation sums of the triple divisor function.