The variance of a general class of multiplicative functions in short intervals

Abstract

We study a general class of multiplicative functions by establishing a connection between their ``short averages" and ``long average". More precisely, we employ Fourier analysis and the counting of rational points on specific binary forms to provide asymptotic estimates for the variance of this class within short intervals. Our results apply to notable multiplicative functions such as $μ_k(n)$, $\frac{ϕ(n)}{n}$, $σ_α(n)$, among others, yielding several new results and improvements in the realm of short interval analysis. Remarkably, our results disprove a conjecture of van Overbeeke concerning the variance of $\frac{ϕ(n)}{n}$.