On the divisor sum function

V.C. Harris, M.V Subbarao
1985 Rocky Mountain Journal of Mathematics  
In memory of Robert A. Smith and Ernst G. Straus 1. Introduction. The function T k (n) representing the number of ways of expressing n as a product of A: factors (the order of the factors being taken into account) has been studied since the time of Dirichlet. In contrast to this well-established function, the corresponding sum function a(n, &), which we define as the sum of the divisors corresponding to such factorizations of«, does not seem to have appeared in the literature. Indeed the only
more » ... ference the authors can submit is their preliminary report [9] . We here formally define the divisor sum function a r (n, k) for the rth powers of these divisors and obtain some identities (including two of a well-known Ramanujan type), and as an application obtain an asymptotic estimate for 2 n ^ a a (n, 3)a b (n, 3) which may be new. We extend the definition of a r {n, k) to the case when k is complex and obtain some asymptotic estimates for its summatory function. Towards the end, we introduce the notation of A>ply perfect numbers and raise some open problems. Preliminaries. Let for k a positive integer, so that r*(«) denotes the number of ways of expressing « as a product of k factors, the order of the factors being taken into account. In particular, let d\dz=n It is clear that if Z^s) stands for the Riemann zeta function, we have W = f;^4^ = <7 + iUa> 1. z k (n) is multiplicative in «, and if n = p%i pp ... pfr
doi:10.1216/rmj-1985-15-2-399 fatcat:t5kwj4myzzfibosd47acwp2bgu