On the residue class distribution of the number of prime divisors of an integer

TL;DR

This paper investigates the distribution of the number of prime divisors of integers across residue classes, generalizing the Liouville function and establishing a weak equidistribution result with explicit error bounds.

Contribution

It introduces a generalized Liouville function using roots of unity and proves a weak equidistribution theorem for the count of integers with a given number of prime divisors modulo m.

Findings

Establishes an asymptotic formula for the distribution of (n) modulo m.

Provides explicit error bounds for the distribution approximation.

Shows the error term cannot be smaller than a certain bound for m > 2.

Abstract

The {\em Liouville function} is defined by $\gl(n):=(-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime divisors of $n$ counting multiplicity. Let $\z_m:=e^{2\pi i/m}$ be a primitive $m$--th root of unity. As a generalization of Liouville's function, we study the functions $\gl_{m,k}(n):=\z_m^{k\Omega(n)}$. Using properties of these functions, we give a weak equidistribution result for $\Omega(n)$ among residue classes. More formally, we show that for any positive integer $m$, there exists an $A>0$ such that for all $j=0,1,...,m-1,$ we have $$#\{n\leq x:\Omega(n)\equiv j (\bmod m)\}=\frac{x}{m}+O(\frac{x}{\log^A x}).$$ Best possible error terms are also discussed. In particular, we show that for $m>2$ the error term is not $o(x^\ga)$ for any $\ga<1$.