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 \( \alpha ^2 \in {\Bbb Q} \).