zbMATH — the first resource for mathematics

The integral mean of the discrepancy of the sequence $$(n\alpha)$$. (English) Zbl 0972.11067
Let $$D_N(\alpha)$$ be the discrepancy of the sequence $$(n\alpha)_{n=1}^N$$ modulo 1, where $$\alpha$$ is a given real number. Then it is proved that $\lim_{N\to\infty} \frac{1} {\log^2 N} \int_0^1 D_N(\alpha) d\alpha= \frac{1}{\pi^2}.$
Reviewer: R.F.Tichy (Graz)

MSC:
 11K38 Irregularities of distribution, discrepancy
discrepancy
Full Text: