On partial derivatives of some summatory functions

TL;DR

This paper explores the use of saddle-point estimates to analyze the distribution of certain integer properties, such as friable integers and squarefree kernels, providing new insights into their frequency and behavior.

Contribution

It introduces novel applications of saddle-point methods to evaluate the distribution of specific summatory functions related to integer properties.

Findings

Saddle-point estimates effectively evaluate the frequency of integers with certain properties.

Revisits Dickman's work on friable integers with new analytical techniques.

Analyzes the distribution of squarefree kernels using advanced asymptotic methods.

Abstract

Let $f$ be a real arithmetic function and let $g:[1,\infty[\to{\mathbb R}$ be a smooth function. We describe two emblematic instances in which saddle-point estimates may be used to evaluate the frequency, on the set of integers $n\leqslant x$, of the event $\{f(n)\leqslant g(n)\}$ from those relevant to the event $\{f(n)\leqslant y\}$. The first example revisits Dickman's historical contribution to the theory of friable integers. The second is concerned with the distribution of the squarefree kernel of an integer.