Abstract :
For the Dirichlet series ∑∞n = 1 (∏kr = 1 σar(n)/ns), we obtain the representation [formula] where K = 2k and br′s are the sums ∑kr = 1 δrar, δr = 0, 1, and ƒk(s) has an Euler product converging in a bigger domain than the domain of convergence of the left side series. The case k = 2 of this is an identity of Ramanujan, with ƒ2(s) = ζ−1(2s − a1 − a2). We also deal with the sum ∑n ≤ x σk(n) and obtain ∑n ≤ x σk(n) = ckxk + 1 + Ek(x) with an explicit ck and Ek(x) = O(xk logk − 1/3x), the O-constant depending only on k. We have obtained the asymptotic estimate for the sum ∑m < MEk(m) as well.
