zeta(3) and Bernoulli

[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.

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

Bests,
Pieter Moree