Raymond Ayoub

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.

On the group ring of a finite abelian group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G . It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

On a theorem of Müller and Carlitz

Let r(n) denote the number of representations of n as a sum of two squares. We have ∑n⩽χ r(n) log(x/n) = Ax +B log x + C + O(x−14) where A, B, C are constants. The evaluation of C in “closed form” is established by a new method in this paper.

Partial triumph or total failure

ConclusionThese examples show that no conclusions can be drawn about the significance of a false proof even when it comes from the work of first class mathematicians.Admittedly, Euler had the advantage that his work was studied by Gauss who, like Rumplestiltskin, could turn straw into gold and Stieltjes lived at the beginning of a period when interest in the Riemann zeta functions and the distribution of primes was high. Perhaps in the final analysis, the significance of an idea depends on the imagination and creativity of those dealing with it.