Hardy-Ramanujan Journal |

When $a$ is an odd positive integer it is implicit in the work of Jacobi that the functions $Y=\sum_1^{\infty} \sigma_a(n)X^n$ where $\sigma_a (n) = \sum_{d/n} d^a$ (the sum of the $a$th powers of the divisors of $n$) satisfy an algebraic differential equation; i.e., there is a polynomial $T$ not identically $0$, such that $T(X, Y, Y_1, \ldots, Y_m)=0$. When $a=1$ we give a new argument based on Ramanujan that we may take $m= 3$ here.

Source : oai:HAL:hal-01104327v1

Volume: Volume 7 - 1984

Published on: January 1, 1984

Imported on: March 3, 2015

Keywords: algebraic differential equation,[MATH] Mathematics [math]

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