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

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 −δ , where δ 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 −δ 0 , where δ0, 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 wellbehaved 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.

Dynamics of $\mathscr{B}$-free systems generated by Behrend sets. I

“…This result relies upon a convolution identity. See [29] and [37] for further details. A weaker result, but sufficient for the estimation in (50), is that q≤n µ 2 (q) ϕ(q) > log n; see [25,Eq.…”