From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Power Series Involving Divisors
Date: 27 Oct 1999 22:58:19 GMT
Newsgroups: sci.math
Keywords: Dirichlet series related to zeta function and arithmetic functions

In article <fqssjqp99bfz@forum.swarthmore.edu>,
Leroy Quet <qqquet@hotbot.com> wrote:
>Are there closed forms to the sums
>sum_{m=1}^infinity [d(m) x^m]=
>sum_{m=1}^infinity [x^m /(1-x^m)]
>and
>sum_{m=1}^infinity [sigma(m) x^m]=
>sum_{m=1}^infinity [x^m /(1-x^m)^2],
>where d(m) is the number of positive divisors of m, and sigma(m) is
>the sum of the positive divisors of m?

Maybe you need to branch out a little and think about sums of other
types, e.g. not Taylor series but Dirichlet series:

 sum_{m=1}^infinity [d(m) m^(-s)]= (zeta(s))^2

 sum_{m=1}^infinity [sigma(m) m^(-s)]= (zeta(s))*(zeta(s-1))

where  zeta   is the Riemann zeta function. There are more of this type:

 sum_{m=1}^infinity [phi(m) m^(-s)]= (zeta(s-1))/zeta(s)

 sum_{m=1}^infinity [mu(m) m^(-s)]= 1/(zeta(s))

 sum_{m=1}^infinity [Lambda(m) m^(-s)]= (zeta'(s))/(zeta(s))

where  phi  is Euler's function, mu is Moebius's, and  Lambda is von Mangoldt's
(Lambda(n)=log(p) if  n  is a power of a prime  p, and is zero otherwise).

Whoa! Six mathematicians named in one short post. Well, seven if you include

dave