New explicit bounds for Mertens function and the reciprocal of the Riemann zeta-function

TL;DR

This paper derives new explicit bounds for the Mertens function by relating it to the zeros of the Riemann zeta-function and establishing bounds for its reciprocal, providing the first explicit results of their kind.

Contribution

It introduces novel explicit bounds for the Mertens function and the reciprocal of the Riemann zeta-function, improving understanding of their behavior.

Findings

New explicit bounds for M(x) involving exponential decay

First explicit bounds for 1/ζ(σ + it) in terms of log t

Comparison of M(x) with zero-sum over ζ(s) zeros

Abstract

In this paper, we establish new explicit bounds for the Mertens function $M(x)$. In particular, we compare $M(x)$ against a short-sum over the non-trivial zeros of the Riemann zeta-function $\zeta(s)$, whose difference we can bound using recent computations and explicit bounds for the reciprocal of $\zeta(s)$. Using this relationship, we are able to prove explicit versions of $M(x) \ll x\exp\left(-\eta_1 \sqrt{\log{x}}\right)$ and $M(x) \ll x\exp\left(-\eta_2 (\log{x})^{3/5} (\log\log{x})^{-1/5}\right)$ for some $\eta_i > 0$. Our bounds with the latter form are the first explicit results of their kind. In the process of proving these, we establish another novel result, namely explicit bounds of the form $1/\zeta(\sigma + it) \ll (\log{t})^{2/3} (\log\log{t})^{1/4}$.