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 $\mu_k(n),\frac{\phi(n)}{n}, \frac{n}{\phi(n)},$ $\mu^2(n)\frac{\phi(n)}{n}$, $\sigma_{\alpha}(n)$, $(-1)^{\#\{p\,: \, p^k|n\}}$ and many others that establish various new results and improvements in short intervals to the literature.