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

Let sigma(k)(n) denote the sum of the k-th powers of the positive divisors of n. Erdos and Kac conjectured that the sum alpha(k) = Sigma(infinity)(n=1) sigma(k)(n)/n! is 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, alpha(1), ..., alpha(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. (C) 2011 Elsevier Inc. All rights reserved