Explicit averages of non-negative multiplicative functions: going beyond the main term
Volume 147 / 2017
Abstract
We produce an explicit formula for averages of the type $\sum _{d\le D}(g\star \mathbf1 )(d)/d$, where $\star $ is the Dirichlet convolution and $g$ a function that vanishes at infinity (more precise conditions are needed, a typical example of an acceptable function is $g(m)=\mu (m)/m$). This formula enables one to exploit the changes of sign of $g(m)$. We use this formula for the classical family of sieve-related functions $G_q(D)=\sum _{{d\le D, (d,q)=1}}{\mu ^2(d)/\varphi (d)}$ for an integer parameter $q$, improving noticeably on earlier results. The remainder of the paper deals with the special case $q=1$ to show how to practically exploit the changes of sign of the Möbius function. It is proven in particular that $|G_1(D)-\log D-c_0|\le 4/\sqrt {D}$ and $|G_1(D)-\log D-c_0|\le 18.4/(\sqrt {D}\log D)$ when $D \gt 1$, for a suitable constant $c_0$.
