Partial sums of typical multiplicative functions over short moving intervals

Abstract

We prove that the k-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval (x, x + H ] matches the corresponding Gaussian moment, as long as H ≪ x/(log x) 2k 2 +2+o( 1) and H tends to infinity with x.We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals (x, x + H ] with H ≪ X/(log X ) W (X ) tending to infinity with X , where x is uniformly chosen from {1, 2, . . ., X }, and W (X ) tends to infinity with X arbitrarily slowly.This makes some initial progress on a recent question of Harper.

Locations

• Algebra & Number Theory - View - PDF
• arXiv (Cornell University) - View - PDF