On sequences involving primes

In this paper we give the set of distribution functions of $f(p_n)$ $\bmod 1$ for a special class of functions $f(x)$. Also, we generalize a result on distribution functions shown by O. Strauch and O. Bla eková to a multi-dimensional case. Applying it, we prove that every uniform distributed sequence $x_n$ $\bmod 1$ and $\log p_n$ $\bmod 1$ is statistically independent. By combining this and a result of G. Rauzy we derive that the sequences $p_n\theta +\log p_n$ and $p_n\theta + p_n/n$ are uniformly distributed $\bmod 1$ with irrational $\theta$.