The algebraic independence of the sum of divisors functions
Let \(\sigma_j(n)=\sum_{d|n}d^j\) be the sum of divisors function, and let \(I\) be the identity function. The author shows that the set of functions \(\{\sigma_i\}_{i=0}^\infty\cup \{I\}\) is algebraically independent. Let \(\pi\) be a prime, \(s\) be an integer relatively prime to \(\pi\), \(\alpha\) be a nonnegative integer, \(r=\pi^\alpha s\) and \(n=\pi^{2\alpha+1}s\), and \(j\) be a fixed, nonnegative integer. The author proves that the identity \[ n^j\sigma_j(r)+r^j\sigma_j(r)=r^j\sigma_j(n) \] is the unique algebraic identity in the variables \(n,r,\sigma_j(r)\), and \(\sigma_j(n)\), up to multiplication by a constant. Furthermore, he proves that any perfect number \(n\) must have the form \(n=\frac{r\sigma(r)}{2r-\sigma(r)}\), with some restrictions on \(r\).
scientific article; zbMATH DE number 5802573