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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

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics