Uniform distribution of some special sequences (Q1063059)

The authors prove the uniform distribution modulo 1 of the sequence (\(\alpha\) f(p)), where \(\alpha\) \(\neq 0\) is a real constant, f(x) is a continuously differentiable function with f(n)/(log n)\({}^{\ell}\to \infty\) (n\(\to \infty)\) for some \(\ell >1\), p runs over the prime numbers, and f satisfies one of the following conditions. (1) f(x)\(\to \infty\) (x\(\to \infty)\), f'(x) log x monotone, n \(| f'(n)| \to \infty\) (n\(\to \infty).\) (2) \(f'(x)>0\), \(f''(x)>0\), \(x^ 2 f''(x)\to \infty\) (x\(\to \infty)\). Furthermore the estimate \(D_ N\ll N^{-1/2+\epsilon}\) has been proved, where \(D_ N\) denotes the discrepancy of the sequence (log n!), \(n=1,2,...,N\) and any \(\epsilon >0\).