Explicit averages of square-free supported functions: to the edge of the convolution method

Sebastian Zuniga Alterman Colloquium Mathematicum MSC: Primary 11N37; Secondary 11A25, 11A41. DOI: 10.4064/cm8337-11-2020 Published online: 14 June 2021

Abstract

We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular behavior on the prime numbers and observe how the nature of this method gives error estimations of order $X^{-\delta }$, where $\delta $ belongs to an open set $I$ of positive reals. In order to have a better error estimation, a natural question is whether or not we can achieve an error term of critical order $X^{-\delta _0}$, where $\delta _0$, the critical exponent, is the right endpoint of $I$. We answer this in the affirmative by presenting a new method that improves qualitatively almost all instances of the convolution method under some regularity conditions; now, the asymptotic estimation of averages of well-behaved square-free supported arithmetic functions can be given with its critical exponent and a reasonable explicit error constant. We illustrate this new method by analyzing a particular average related to the work of Ramaré–Akhilesh (2017), which leads to notable improvements when imposing non-trivial coprimality conditions.

Authors

  • Sebastian Zuniga AltermanInstitut de Mathématiques de Jussieu
    Université Paris Diderot P7
    Bâtiment Sophie Germain, 8 Place Aurélie Nemours
    75013 Paris, France
    e-mail

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