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

Abstract

IWe prove new unconditional and explicit upper bounds for the reciprocal of the Riemann zeta function in the critical strip, of the orders $(\log t)^{11/12}$, $(\log t)^{11/12}(\log\log t)^{-3/4}$, and $(\log t)^{2/3}(\log\log t)^{1/4}$; these hold in prescribed classical, Littlewood, and Korobov--Vinogradov zero-free regions, respectively. From these bounds, we also derive new unconditional and explicit upper bounds for the Mertens function $M(x)$, of the orders $x (\log x)\exp\!\left(-\eta_1 \sqrt{\log x}\right)$, $x (\log x) \exp\!\left(-\eta_2 \sqrt{\log x \log\log x}\right)$, and $x (\log x)\exp\!\left(-\eta_3 (\log x)^{3/5}(\log\log x)^{-1/5}\right)$, for suitable constants $\eta_1,\eta_2,\eta_3>0$. A key feature of our approach is the use of computational smoothing, which ensures the resulting numerical bounds are strong.