On the QQ-linear independence of the sums ∑n=1∞σk(n)/n!

Let σk(n)σk(n) denote the sum of the k-th powers of the positive divisors of n. Erdős and Kac conjectured that the sumαk=∑n=1∞σk(n)n! is irrational for k⩾1k⩾1. This is known to be true for k=1k=1, 2 and 3. Fix r⩾1r⩾1. In this article we give a precise criterion for 1,α1,…,αr1,α1,…,αr to be QQ-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=50r=50.