Sums of fractional parts and sum of restricted divisors of a number

Abstract

Let us consider a strictly increasing sequence of positive integers a_n such that A(x) is the distribution function of the sequence. That is, A(x)=∑ _{an≤ x} 1. We study the sum ∑ _{an≤ x} a_n^s{x / a_n} and apply this formula in the study of the sum of a-divisors of a number. The distribution functions A(x) considered are very general. The methods used are very elementary.