On the distribution of the strongly multiplicative function $2^{ω(n)}$ on the set of natural numbers
K. Venkatasubbareddy, A. Sankaranarayanan
TL;DR
The paper studies the distribution of the strongly multiplicative function $f(n)=2^{\omega(n)}$ by analyzing its Dirichlet series $F(s)=\zeta^2(s)/\zeta(2s)$. Using Perron's formula and contour integration, it derives conditional asymptotics under the strong Riemann hypothesis that place zeros of $\zeta(2s)$ on $\Re(s)=\tfrac{1}{4}$ and yield oscillatory main terms from these zeros in addition to the main terms $A_1 x \log x$ and $A_2 x$, with $A_1=1/\zeta(2)$ and $A_2=(2\gamma-1)/\zeta(2)$. Unconditionally, the authors account for possible cancellations and obtain a second result that includes residues from zeros of $\zeta(s)$ and an error term $O(x^{8/29+\varepsilon})$, which is sharper than the Dirichlet divisor problem exponent $131/416$. The work clarifies how RH affects the fine structure of the sum $\sum_{n\le x} 2^{\omega(n)}$ and suggests extensions to related arithmetic sums via similar L-function factorizations.
Abstract
In this paper, we study the distribution of the sequence of integers $2^{ω(n)}$ under the assumption of the strong Riemann hypothesis, where $ω(n)$ denotes the number of distinct prime divisors of $n$. We provide an asymptotic formula for the sum $\displaystyle\sum_{n\leq x}2^{ω(n)}$ under this assumption. We study the sum $\displaystyle\sum_{n\leq x}2^{ω(n)}$ unconditionally too.