Remarques sur une somme liée à la fonction de Möbius

Abstract

For integer $n⩾1$ and real number $z⩾1$, define $M(n,z):={\sum }_{d|n,d⩽z}\mu (d)$ where $\mu$ denotes the Möbius function. Put $\mathcal{L}(y):=\exp \left\{(\log y{)}^{3/5}/({\log }_{2}y{)}^{1/5}\right\}$ $(y⩾3)$. We show that, for a suitable, explicit, constant $L>0$ and some absolute $c>0$, we have $S(x,z)=Lx+O\left(x/\mathcal{L}(3\xi {)}^c\right)$ uniformly for $x⩾1$, $\xi ⩽z⩽x/\xi$.

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