Almost uniform distribution modulo 1 and the distribution of primes Summary: Almost uniform distribution modulo 1 and the distribution of primes Shigeki Akiyama Abstract Let (an), n = 1, 2, . . . be a sequence of real numbers which is related with number theoretic functions such as Pn, the n-th prime. We study the distribution of the fractional parts of (an) using the concept of "almost uniform distribution" defined in [9]. Then we can show a generalization of the results of [2] on the convex property of log Pn. The method may be extended as well to other oscillation problems of number theoretical interest. Let (an), n = 1, 2, . . . be a sequence of real numbers and A(I, (an), N) be the counting function, that is, the number of n = 1, 2, . . . , N that {an} is contained in a certain interval I [0, 1]. Here we denote by {an} = an - [an], the fractional part of an. First we recall a kind of generalization of the classical definition of uniform distribution modulo 1 (see [9], [3] and [8]). Definition. The sequence (an) is said to be almost uniformly distributed modulo 1 if there exist a strictly increasing sequence of natural numbers (nj), j = 1, 2, . . . and, for every pair of a, b with 0 a < b 1, lim j Collections: Mathematics