Toward optimal exponent pairs

TL;DR

This paper introduces a framework to identify optimal exponent pairs for various number-theoretic problems, leading to progress on bounds for the Riemann zeta-function, moments estimation, and divisor problems.

Contribution

It develops a systematic method to compute optimal exponent pairs for arbitrary objectives, advancing solutions to several open problems in analytic number theory.

Findings

Improved bounds for the Riemann zeta-function in the critical strip

Enhanced estimates of moments of ta(1/2 + it)

Progress on the generalized Dirichlet divisor problem

Abstract

We quantify the set of known exponent pairs $(k, \ell)$ and develop a framework to compute the optimal exponent pair for an arbitrary objective function. Applying this methodology, we make progress on several open problems, including bounds of the Riemann zeta-function $\zeta(s)$ in the critical strip, estimates of the moments of $\zeta(1/2 + it)$ and the generalised Dirichlet divisor problem.