On the sum of Divisors Function

Abstract For the Dirichlet series ∑ ∞ n = 1 (∏ k r = 1 σ a r ( n )/ n s ), we obtain the representation [formula] where K = 2 k and b r ′s are the sums ∑ k r = 1 δ r a r , δ r = 0, 1, and ƒ k ( 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 ƒ 2 ( s ) = ζ −1 (2 s − a 1 − a 2 ). We also deal with the sum ∑ n ≤ x σ k ( n ) and obtain ∑ n ≤ x σ k ( n ) = c k x k + 1 + E k ( x ) with an explicit c k and E k ( x ) = O ( x k log k − 1/3 x ), the O -constant depending only on k . We have obtained the asymptotic estimate for the sum ∑ m M E k ( m ) as well.