An observation on the sums of divisors

Abstract

Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that ${\prod }_{n=1}^{\infty }(1-x^n)={\sum }_{n=-\infty }^{\infty }(-1{)}^n{x}^{n(3n-1)/2}$. This paper assumes the pentagonal number theorem and uses it to prove a recurrence relation for the sum of divisors function. The term "pentagonal numbers" comes from polygonal numbers. Euler takes the logarithmic derivative of both sides. Then after multiplying both sides by $-x$, the left side is equal to ${\sum }_{n=1}^{\infty }\sigma (n)x^n$, where $\sigma (n)$ is the sum of the divisors of $n$, e.g. $\sigma (6)=12$. This then leads to the recurrence relation for $\sigma (n)$. I have been studying in detail all of Euler's work on the pentagonal number theorem, and more generally infinite products. I would be particularly interested to see if anyone else worked with products and series like these between Euler and Jacobi, and I would enjoy hearing from anyone who knows something about this.

13 pages; E243