Sur le biais d'une loi de probabilit\'e relative aux entiers friables

TL;DR

This paper investigates the bias in the probability distribution on y-friable integers, proposing a measure for this bias and demonstrating a Gaussian distribution related to it.

Contribution

It introduces a quantitative measure of the bias in the probability law on y-friable integers and establishes a related Gaussian distribution.

Findings

Identifies a structural bias in the probability law on y-friable integers.

Proposes a quantitative measure for the bias.

Shows a Gaussian distribution related to the bias.

Abstract

The standard probability law on the set $S(x,y)$ of $y$-friable integers not exceeding $x$ assigns to each friable integer $n$ a probability proportional to $1/n^\alpha$ where $\alpha=\alpha(x,y)$ is the saddle-point of the inverse Laplace integral for $\Psi(x,y):=|S(x,y)|$. This law presents a structural bias inasmuch it weights integers $>x$. We propose a quantitative measure of this bias and exhibit a related Gaussian distribution.