A lower bound for the variance in arithmetic progressions of some multiplicative functions close to $1$

TL;DR

This paper establishes lower bounds for the variance in arithmetic progressions of multiplicative functions close to 1, including divisor functions and smooth number indicators, enhancing existing results and connecting to classical theorems.

Contribution

It provides new lower bounds for variance in arithmetic progressions of specific multiplicative functions, strengthening prior results and extending to prime factor counting functions.

Findings

Lower bounds for variance of divisor functions near 1

Matching bounds for smooth number indicator functions

Lower bounds for prime factors counting functions

Abstract

We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to $1$. Specifically, we consider $\alpha_N$-fold divisor functions, when $\alpha_N$ is a sequence of positive real numbers approaching $1$ in a suitable way or $\alpha_N=1$, and the indicator of $y$-smooth numbers, for suitably large parameters $y$. As a corollary, we will strengthen a previous author's result on the first subject and obtain matching lower bounds to some Barban-Davenport-Halberstam type theorems for $y$-smooth numbers. Incidentally, we will also find a lower bound for the variance in arithmetic progressions of the prime factors counting functions $\omega(n)$ and $\Omega(n)$.