zeta(3) and Bernoulli numbers

[Pieter Moree]

Dear seqfans,

Let A=\sum_{k=1}^{\infty}B_{2k}\pi^{2k}/(2*k+3)!

In his book `The development of prime number theory' at p. 134
Prof. Narkiewicz states a formula of J.C. Kluyver (1896):

\zeta(3)=4*Pi*(1/6-A)/5.

Narkiewicz defines B_n by z/(e^z-1)=\sum_{n=0}^{\infty}B_n*z^n/n!

With this definition the formula is certainly not correct. I
tried some older definitions of Bernoulli numbers, but could not
get the formula matching.

I am looking for a modification of this formula with
`least Hamming distance' so that it matches. (The original
paper of Kluver is in Bull. Sci. Math. (1896) 20, 116-119, something not
available to me. If somebody could send me a copy of this I would be
very grateful.)

A formula similar to Kluyver's for zeta(3) does exist. It is due
to Euler (1772) and was rediscovered by various people (it uses
Pi^2 rather than Pi).

Kluyver was the first Dutch analytic number theorist. His PhD students
include Kloosterman and Van der Corput.

Bests,
Pieter Moree