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

2022/08/28 by Darbar, Pranendu, Das, Mithun Kumar
#11K65 #11M06 #11N37 #42B10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2208.13238

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 μ_k(n),(ϕ(n))/(n), (n)/(ϕ(n)), μ²(n)(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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