 mersenneforum.org > Math Why zeta(-2n)=0 ?
 Register FAQ Search Today's Posts Mark Forums Read 2005-12-07, 19:47 #1 Crook   Nov 2004 100002 Posts Why zeta(-2n)=0 ? I know this wouldn't even deserve to be posted, but could someone explain me why zeta(-2n)=0 ? Regards.   2005-12-08, 02:34   #2
John Renze

Nov 2005

3016 Posts Quote:
 Originally Posted by Crook I know this wouldn't even deserve to be posted, but could someone explain me why zeta(-2n)=0 ? Regards.
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 http://eww2lx.wri.wolfram.com/RiemannZetaFunction.html 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,
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