 
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 nth 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
