Almost uniform distribution modulo 1 and the distribution of primes (Q1271914)

A sequence \((a_n)\) of real numbers is said to be almost uniformly distributed \(\text{mod }1\) if there exists a strictly increasing sequence of natural numbers \((n_j)\) such that for every pair of \(a\), \(b\) with \(0\leq a< b\leq 1\) we have \[ \lim_{j\to\infty} \frac 1{n_j}\# \{n\leq N:a\leq a_n-[a_n]<b\}=b-a. \] The author states without proofs some properties of such sequences which generalize classical results on uniform distribution mod 1 (e.g. Fejér's theorem and Weyl's criterion). Then he applies them to show that the sequence \(\log p_n\), where \(p_n\) denotes the \(n\)-th prime, is not almost uniformly distributed mod 1 (Theorem 2) and that \(\Delta^2\log p_n=\log p_{n+2}-2\log p_{n+1}+\log p_n\) changes sign infinitely many times as \(n\) tends to infinity (Theorem 3). Some generalizations of these results are also proved.