Euler zeta function
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000:
GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)

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)



Chris K. Caldwell © 1999-2019 (all rights reserved)