  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.
