Euler zeta function

Euler studied the sum
zeta(s) = 1 +
2^(-s) + 3^(-s) + 4^(-s)...
for integers s > 1 (clearly zeta(1) is infinite).  Euler discovered a formula relating zeta(2k) to the Bernoulli numbers yielding results such as zeta(2)=pi^2/6 and zeta(4)=pi^4/90.

What has this got to do with the primes?  The answer is in the following product taken over the primes p (also discovered by Euler):

[Euler's Prod]
Euler wrote this as
[Euler's Prod]

See Also: RiemannZetaFunction

Related pages (outside of this work)

Printed from the PrimePages <> © Chris Caldwell.