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

Volume 147 / 2017

O. Ramaré, P. Akhilesh Colloquium Mathematicum 147 (2017), 275-313 MSC: Primary 11N37, Secondary 11N36, 11N56. DOI: 10.4064/cm6080-4-2016 Published online: 31 January 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 $c_0$.

Authors

  • O. RamaréCNRS / Institut de Mathématiques de Marseille
    Aix Marseille Université, U.M.R. 7373
    Site Sud, Campus de Luminy, Case 907
    13288 Marseille Cedex 9, France
    e-mail
  • P. AkhileshHarish-Chandra Research Institute
    Chhatnag Road
    Jhusi, Allahabad 211 019, India
    e-mail

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