2005/06/20 by Leonhard Euler, Euler, Leonhard
Mathematics · #01A50 #11-03 #11B68 #11M06 #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.HO #math.NT #msc:01A50 #msc:11-03 #msc:11B68 #msc:11M06
paper · pdf · doi:10.48550/arxiv.math/0506415
8 pages, 1 figure
openalex publication_date 2005/06/20 · arxiv created 2008/02/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This translation has been published in Stephen Hawking (ed.), "God Created the Integers", published in 2007 by Running Press. There may have been some changes to the final published version and this copy. This is a translation from the Latin original, "De summis serierum reciprocarum" (1735). E41 in the Enestrom index. In this paper Euler finds an exact expression for the sum of the squares of the reciprocals of the positive integers, namely pi2/6. He shows this by applying Newton's identities relating the roots and coefficients of polynomials to the power series of the sine function. Indeed, in other words this result is zeta(2)=pi2/6, and Euler also works out zeta(4),zeta(6),...,zeta(12). His method will work out zeta(2n) for all n, but he does not give a general expression for zeta(2n); he gives a general expression involving the Bernoulli numbers in a latter paper.