On the Q-linear independence of the sums ∑n=1∞σk(

Abstract

Let σ k (n) denote the sum of the k-th powers of the positive divisors of n. Erdős and Kac conjectured that the sumis irrational for k 1. This is known to be true for k = 1, 2 and 3.Fix r 1. In this article we give a precise criterion for 1, α 1 , . . . , α r to be Q-linearly independent, assuming a standard conjecture of Schinzel on the prime values taken by a family of polynomials. We have verified our criterion for r = 50.