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

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 $μ_k(n),\frac{ϕ(n)}{n}, \frac{n}{ϕ(n)},$ $μ^2(n)\frac{ϕ(n)}{n}$, $σ_α(n)$, $(-1)^{\#\{p\,: \, p^k|n\}}$ and many others that establish various new results and improvements in short intervals to the literature.

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