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

TL;DR

This paper analyzes the variance of a broad class of multiplicative functions in short intervals, connecting short and long averages using Fourier analysis and rational point counting, leading to new insights and results.

Contribution

It introduces a method to estimate the variance of various multiplicative functions in short intervals, extending previous results with new techniques and applications.

Findings

Asymptotic estimates for variance in short intervals

Application to multiple multiplicative functions including μ_k, φ/n, and σ_α

Improved results in short interval analysis

Abstract

We study a general class of multiplicative functions by relating "short averages" to its "long average". More precisely, we estimate asymptotically the variance of such a class of functions in short intervals using Fourier analysis and counting rational points on certain binary forms. Our result is applicable to the interesting multiplicative functions $\mu_k(n),\frac{\phi(n)}{n}, \frac{n}{\phi(n)},$ $\mu^2(n)\frac{\phi(n)}{n}$, $\sigma_{\alpha}(n)$, $(-1)^{\#\{p\,: \, p^k|n\}}$ and many others that establish various new results and improvements in short intervals to the literature.