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