Euler and the Zeta Function

As we saw on Wednesday, in E-19, Euler discovered the Gamma function while trying to “interpolate the hypergeometric series.” Then, in E-20, similar efforts with the harmonic series enabled him to approximate ζ(2). Here, we skip over E-25 where he discovers Euler-Maclaurin series, E-41, where he got the exact value for ζ(2), and E-47 where he begins his study of γ, the Euler-Mascheroni constant. We go to E-72, the paper Bill Dunham calls the first paper in analytic number theory, where Euler discovers the basic properties of the “Riemann” Zeta function. At the end of E-20, Euler notes two intractable series 1 1 1 1 1 ... 3 7 15 2 1 n + + + + + − and 1 1 1 1 1 1 . 3 7 8 15 24 26 etc + + + + + + Skip over several related papers E-25 Euler-Maclaurin series E-41 Basel problem – exact value for ζ(2) E-47 γ, the Euler-Mascheroni constant E-41 exact value of ζ(2) E-47 Euler-Mascheroni constant E-72 Product-sum formula for Zeta function in the paper Dunham calls the beginning of analytic number theory (1737) E-72 Various observations on infinite series Euler begins with one of those “intractable” series: 1 1 1 1 1 1 ... 1 3 7 8 15 24 26 + + + + + + = Theorem 1: This infinite series 1 3 1 7 1 8 1 15 1 24 1 26 1 31 1 35 + + + + + + + + etc. , with denominators one less than all those numbers which are powers, either second powers or higher powers, of ordinary whole numbers, and whose general term can be expressed by the formula 1 1 m − , where m and n denote whole numbers greater than one, the sum of this whole series is =1.