Explicit averages of square-free supported functions: to the edge of the convolution method
Volume 168 / 2022
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.
