A Generalization of a Result of Hardy and Littlewood

Details A Generalization of a Result of Hardy and Littlewood A Generalization of a Result of Hardy and Littlewood

2007-09-18

arxiv.org/abs/0709.2882v2

Mathematics

Number Theory

11 pages

Scientific paper

In this note we study the growth of \sum_{m=1}^M\frac1{\|m\alpha\|} as a function of M for different classes of \alpha\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 \alpha. For a non-decreasing function \phi tending to infinity, \limsup_{M\to\infty}\frac1{\phi(\log M)}\bigg[\frac1{M\log M}\sum_{m=1}^M\frac1{\|m\alpha\|}\bigg] is zero or infinity according as \sum\frac1{k\phi(k)} converges or diverges.

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