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

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 $1/\zeta(\sigma + it) \ll (\log{t})^{2/3} (\log\log{t})^{1/4}$.