Conversions explicites entre des fonctions sommatoires de la fonction de Möbius
Florian Daval
TL;DR
This paper develops explicit, real-analysis–based methods to translate explicit bounds for the summatory Möbius function $M(x)$ into bounds for the arithmetic sum $m(x)=\sum_{n\le x} μ(n)/n$, and conversely. The core contribution is a framework of conversion lemmas using kernels $G_1$ and $H_2$ that express $m_1(x)$ and related quantities as weighted integrals of $M$ (or vice versa), yielding explicit numerical bounds across wide ranges and enabling direct propagation of known explicit $M(x)$-bounds to $m(x)$-bounds. The authors obtain concrete results such as $|m(x)|\le 1/4343$ for $x\ge 2.16\times 10^6$ and $|m_1(x)|\le 1/11{,}470{,}909$ for $x\ge 2.2\times 10^{12}$, and they translate classic explicit bounds for $|M(x)|$ (e.g., $|M(x)|/x\le 0.013/\log x$) into corresponding bounds for $m$ and related checked functions. They also prove an Odlyzko–te Riele type result showing $\varlimsup |m(x)|\sqrt{x} > 1$, reinforcing the established refutation of Mertens’ conjecture and highlighting oscillations of Möbius sums. Collectively, the work provides a rigorous, computable bridge between $M$- and $m$-type bounds, bridging theory and computation in explicit analytic number theory.
Abstract
By using exclusively real analysis, we give explicit estimates of some classical summatory functions involving the Möbius function.