Decoupling, exponential sums and the Riemann zeta function

TL;DR

This paper introduces a novel decoupling inequality for curves that enhances mean value estimates for exponential sums, resulting in a tighter bound on the Riemann zeta function along the critical line.

Contribution

It develops a new decoupling inequality for curves, leading to improved bounds for exponential sums and the Riemann zeta function on the critical line.

Findings

New decoupling inequality for curves

Improved mean value theorem for exponential sums

Enhanced bound |  1/2+it|   t^{53/342+}

Abstract

We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in [H]. In particular, this leads to an improved bound $|\zeta(\frac 12+it)|\ll t^{53/342+\varepsilon}$ for the zeta function on the critical line