Hardy-Ramanujan Journal |
When a is an odd positive integer it is implicit in the work of Jacobi that the functions Y=∑∞1σa(n)Xn where σa(n)=∑d/nda (the sum of the ath powers of the divisors of n) satisfy an algebraic differential equation; i.e., there is a polynomial T not identically 0, such that T(X,Y,Y1,…,Ym)=0. When a=1 we give a new argument based on Ramanujan that we may take m=3 here.