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.
Jacob's ladders, logarithmic modification of the Hardy-Littlewood integral (1918), Titchmarsh's $Ω$-theorem (1928) and new point of contact with the Fermat-Wiles theorem
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relation to the Titchmarsh's sums (1943) and the Fermat-Wiles theorem
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918), the classical Dirichet's sum of divisors (1849) and their relationship with the Fermat-Wiles theorem
Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem
Jacob's ladders, almost exact decomposition of certain increments of the Hardy-Littlewood integral (1918) by means of the Raabe's integral and the thirteenth equivalent of the Fermat-Wiles theorem
Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics
我们建立了$q(x_1,\cdots,x_n)=m$形式的仿射二次型中Zebriki开子集的Hardy-Littlewood性质（à la Borovoi-Rudnick），其中$q$是$n\geqslant 3$变量的非退化整二次型，$m$是非零整数.给出了取互质多项式值的整点密度的渐近公式，这是仿射二次曲面在无穷远处强逼近性质的算术纯性的一个定量形式。
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relations to the Selberg's formula (1946) and the Fermat-Wiles theorem
