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-   -   Why zeta(-2n)=0 ? (https://www.mersenneforum.org/showthread.php?t=5102)

Crook 2005-12-07 19:47

Why zeta(-2n)=0 ?
 
I know this wouldn't even deserve to be posted, but could someone explain me why zeta(-2n)=0 ? Regards.

John Renze 2005-12-08 02:34

[QUOTE=Crook]I know this wouldn't even deserve to be posted, but could someone explain me why zeta(-2n)=0 ? Regards.[/QUOTE]

It's hard to answer this without knowing your mathematical level, but here is one way to look at it.

Start with one version of the functional equation for the zeta function:

zeta(1-s) == 2 (2pi)^(-s) cos(s*pi/2)gamma(s)zeta(s)

See Equation 10 at [url]http://eww2lx.wri.wolfram.com/RiemannZetaFunction.html[/url] or many other sources for this equation.

Substitute s = 2n+1 where n is a positive integer to get

zeta(-2n) == 2 (2pi)^(-s) cos(n*pi + 1/2)(2n+1)!*zeta(2n+1)

cos(n*pi + 1/2) == 0 and zeta(2n+1) is finite, so you get

zeta(-2n) == 0

Hope this helps,
John


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