On limiting distributions of arithmetic functions

TL;DR

This paper investigates the limiting distributions of specific arithmetic functions related to prime exponents, introduces a new probabilistic distribution, and demonstrates its applicability to various arithmetic functions.

Contribution

It introduces a novel discrete distribution dependent on a function f and connects it to the limiting behavior of arithmetic functions like M(n) and m(n).

Findings

Established the second moments and limiting distributions of M(n) and m(n).

Defined a new probabilistic distribution based on a function f in [0,1].

Provided examples of arithmetic functions conforming to this distribution.

Abstract

For a natural number n, let M(n) denote the maximum exponent of any prime power dividing n, and let m(n) denote the minimum exponent of any prime power dividing n. We study the second moments of these arithmetic functions and establish their limiting distributions. We introduce a new discrete probabilistic distribution dependent on a function f taking values in [0,1], study its first two moments, and provide examples of several arithmetic functions satisfying such distribution as their limiting behavior.