Zur Metrik der Diskrepanz

LetI=0,1),dω the product measure onI ^? of the Lebesgue measure onI, D _N (ω) the discrepancy ofω ∈I ^?. We show that for α>0 $$\mathop {\lim }\limits_{N \to \infty } N^{\alpha /2} \mathop \smallint \limits_{I^\mathbb{N} } D_N^\alpha (\omega ) d\omega = 4.2 - (3\alpha /2)(2\alpha - 1 - 1) \zeta (\alpha ) \Gamma (\alpha /2 + 1)$$ Furthermore we sharpen and generalize this result.