2004/11/26 by Leonhard Euler, Euler, Leonhard, Jordan Bell +1
Mathematics · #01A50 #11B37 #FOS: Mathematics #History and Overview (math.HO) #Number Theory (math.NT) #math.HO #math.NT #msc:01A50 #msc:11B37
paper · pdf · doi:10.48550/arxiv.math/0411587
13 pages; E243
arxiv created 2009/07/18 · arxiv updated 2009/12/01
Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that ∏n=1^∞ (1-xn)=∑n=-∞^∞ (-1)n xn(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 ∑n=1^∞ σ(n) xn, where σ(n) is the sum of the divisors of n, e.g. σ(6)=12. This then leads to the recurrence relation for σ(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.