Explicit averages of non-negative multiplicative functions: going beyond the main term

O. Ramaré¹ ; P. Akhilesh²

¹ CNRS / Institut de Mathématiques de Marseille Aix Marseille Université, U.M.R. 7373 Site Sud, Campus de Luminy, Case 907 13288 Marseille Cedex 9, France
² Harish-Chandra Research Institute Chhatnag Road Jhusi, Allahabad 211 019, India

Colloquium Mathematicum, Tome 147 (2017) no. 2, pp. 275-313

Résumé

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 $c_0$.

Détail

DOI : 10.4064/cm6080-4-2016

Keywords: produce explicit formula averages type sum star mathbf where star dirichlet convolution function vanishes infinity precise conditions needed typical example acceptable function formula enables exploit changes sign formula classical family sieve related functions sum varphi integer parameter improving noticeably earlier results remainder paper deals special practically exploit changes sign bius function proven particular log d c sqrt log d c sqrt log suitable constant nbsp