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