Remarques sur une somme li\'ee \`a la fonction de M\"obius

TL;DR

This paper investigates a sum involving the Möbius function and provides an explicit asymptotic formula with a precise error term, valid uniformly over a broad range of parameters.

Contribution

It establishes a new explicit asymptotic approximation for a Möbius sum with a detailed error bound, extending previous results.

Findings

Derived an explicit main term for the sum involving the Möbius function.

Provided a uniform error bound that improves understanding of Möbius sums.

Extended the range of parameters where the asymptotic holds.

Abstract

For integer $n\geqslant 1$ and real number $z\geqslant 1$, define $M(n,z):=\sum_{d|n,\,d\leqslant z}\mu(d)$ where $\mu$ denotes the M\"obius function. Put ${\cal L}(y):=\exp\left\{(\log y)^{3/5}/(\log_2y)^{1/5}\right\}$ $(y\geqslant 3)$. We show that, for a suitable, explicit, constant $L>0$ and some absolute $c>0$, we have $S(x,z)= Lx+O\left({x/{\cal L}(3\xi)^c}\right)$ uniformly for $x\geqslant 1$, $\xi\leqslant z\leqslant x/\xi$.