A Generalization of a Result of Hardy and Littlewood

Abstract

In this note we study the growth of \sum_{m=1}^M\frac1{\|mα\|} as a function of M for different classes of α\in[0,1). Hardy and Littlewood showed that for numbers of bounded type, the sum is \simeq M\log M. We give a very simple proof for it. Further we show the following for generic α. For a non-decreasing function ϕtending to infinity, \limsup_{M\to\infty}\frac1{ϕ(\log M)}\bigg[\frac1{M\log M}\sum_{m=1}^M\frac1{\|mα\|}\bigg] is zero or infinity according as \sum\frac1{kϕ(k)} converges or diverges.

arXiv comment11 pages