DISTRIBUTION OF THE SEQUENCE p /n mod 1

In this paper we show that the sequence pn/n mod 1, n = 1, 2, . . . , where pn is the nth prime and the sequence log n mod 1, n = 1, 2, . . . have the same set of distribution functions. The presented proof is divided into two steps. In Step 1 we derive that pn/n mod 1 has the same set of distribution functions as the sequence log(n log n) mod 1 and in Step 2 we prove that log(n log n) mod 1, has the same set of distribution functions as log n mod 1. Step 2 is based on a new method for computing distribution functions of xn + yn mod 1 from the distribution functions of the two–dimensional sequence (xn, yn) mod 1 by using the Riemann-Stieltjes integration of related two-dimensional functions. We apply this method to xn = log n mod 1 and yn = log log n mod 1, n = 2, 3, . . . . In an al- ternative Step 2 we use some generalization of a result of Koksma on distribution functions.