On the discrepancy of the sequence $ \bigl (\alpha \sqrt {n}\bigr) $ II

Abstract.

Let $\alpha \ge 0$ and denote by $D_N (\alpha)$ the discrepancy of the sequence $(\alpha \sqrt {n})_{n\ge 1}$. Employing a recent result of J.Schoißengeier and the author we calculate $\lim \sup \limits_{N\to \infty} N^{-1/2}D_N(\alpha)$ in the case $\alpha ^2 \in {\Bbb Q}$.