Almost Convergent 0-1-Sequences and Primes

Abstract

We study 0-1-sequences and establish the connection between the values of the upper and lower Sucheston functional on such sequence and the set of all possible divisors of the elements in the sequence support. If the union of the sets of all simple divisors of the elements in a 0-1-sequence support is finite then the sequence is almost convergent to zero. We study the 0-1-sequences whose support consists exactly of the multiples of the elements in a given set, and establish some necessary and sufficient conditions for the upper Sucheston functional to be equal to 1 on such sequence. We prove that there are infinitely many sequences at which the lower Sucheston functional is 1, and the lower Sucheston functional never vanishes at any of such sequences.