Vinogradov's integral and bounds for the Riemann zeta function

TL;DR

This paper establishes improved bounds for the Riemann zeta function in the critical strip using Vinogradov's integral, leading to tighter estimates than previously known.

Contribution

It introduces a novel method linking Vinogradov's integral bounds with zeta function estimates, significantly reducing the exponent in the bounds.

Findings

New explicit bounds for ta(s) in the critical strip

Enhanced bounds for Vinogradov's integral and solutions of incomplete systems

Improved exponent in the zeta function bounds from 18.8 to 4.45

Abstract

We show for all $1/2 \le \sigma \le 1$ and $t\ge 3$ that $\zeta(\sigma+it)| \le 76.2 t^{4.45 (1-\sigma)^{3/2}}$, where $\zeta$ is the Riemann zeta function. This significantly improves the previous bounds, where $4.45$ is replaced by $18.8$. New ingredients include a method of bounding $\zeta(s)$ in terms of bounds for Vinogradov's Integral (aka Vinogradov's Mean Value) together with bounds for "incomplete Vinogradov systems", explicit bounds for Vinogradov's integral which strengthen slightly bounds of Wooley (Mathematika 39 (1992), no. 2, 379-399), and explicit bounds for the count of solutions of "incomplete Vinogradov systems", following ideas of Wooley (J. Reine Angew. Math. 488 (1997), 79-140)