## Uniform distribution of some special sequences.(English)Zbl 0573.10023

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 $$\epsilon >0$$.
Reviewer: D.Leitmann

### MSC:

 11J71 Distribution modulo one 11L03 Trigonometric and exponential sums (general theory)

### Keywords:

uniformly distributed mod 1; log-type function; discrepancy
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### References:

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