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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
Keywords:
discrepancy
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