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

Abstract

Let us consider a strictly increasing sequence of positive integers an such that A(x) is the distribution function of the sequence. That is, A(x)=∑ an≤ x 1. We study the sum ∑ an≤ x ans{x / an} 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.

Sequences related to the e number and Bernoulli numbers

“…The Euler's constant γ appear, for instance, in certain sums of fractional parts [4] [5]. The values of the zeta function ζ(s) also appear, for instance, in certain sums of fractional parts [3] and the counting function of many sequences of positive integers [2]. In this note, we study some sequences related simultaneously with the e number and the Bernoulli numbers.…”

On the equation Π_P Φ_3(P) = Π_P P^2 over F_2[x]

“…This σ function is more natural, but also more complex, than the usual sum of divisor function σ 1 (A) = D|A 2 deg(A) . We consider this function σ as the natural analogue on F 2 [x] of the usual sum of divisors function over the integers Z (see [17]). For instance, some divisors D of A can sum up to 0, while a sum over D of 2 deg D is always greater than 0.…”