# An explicit sub-Weyl bound for $\zeta(1/2 + it)$

## TL;DR

This paper establishes an explicit sub-Weyl bound for the Riemann zeta function on the critical line, providing the sharpest known bounds for large t values, which advances understanding of its growth behavior.

## Contribution

It presents the first explicit sub-Weyl bound for ta(1/2 + it), improving the known upper bounds on its magnitude for large t.

## Key findings

- Proves ta(1/2 + it)  66.7 t^{27/164} for t  3
- Provides the sharpest bounds for ta(1/2 + it) for t  ^{61}
- Enhances understanding of the growth rate of ta on the critical line.

## Abstract

In this article we prove an explicit sub-Weyl bound for the Riemann zeta function $\zeta(s)$ on the critical line $s = 1/2 + it$. In particular, we show that $|\zeta(1/2 + it)| \le 66.7\, t^{27/164}$ for $t \ge 3$. Combined, our results form the sharpest known bounds on $\zeta(1/2 + it)$ for $t \ge \exp(61)$.