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.

We have substantially revised and extended the previous paper, introducing a unified framework that yields classical, Littlewood, and Korobov-Vinogradov type results as special cases. Combined with an updated computational smoothing method, this leads to stronger quantitative and asymptotic results throughout. Comments, corrections, and suggestions are warmly welcomed