On the distribution of the strongly multiplicative function $2^{\omega(n)}$ on the set of natural numbers

TL;DR

This paper investigates the distribution of the multiplicative function 2^{ω(n)} over natural numbers, providing asymptotic formulas both under the strong Riemann hypothesis and unconditionally, enhancing understanding of its behavior.

Contribution

It offers new asymptotic estimates for the sum of 2^{ω(n)} under different assumptions, advancing knowledge of multiplicative functions' distribution.

Findings

Asymptotic formula derived under the strong Riemann hypothesis.

Unconditional asymptotic estimate established.

Insights into the distribution of 2^{ω(n)} across natural numbers.

Abstract

In this paper, we study the distribution of the sequence of integers $2^{\omega(n)}$ under the assumption of the strong Riemann hypothesis, where $\omega(n)$ denotes the number of distinct prime divisors of $n$. We provide an asymptotic formula for the sum $\displaystyle\sum_{n\leq x}2^{\omega(n)}$ under this assumption. We study the sum $\displaystyle\sum_{n\leq x}2^{\omega(n)}$ unconditionally too.