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Topic: Use of functional eqn to show that zeros of Riemann Zeta occur in pair
Replies: 9   Last Post: Aug 31, 2009 10:55 AM

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Dik T. Winter

Posts: 7,899
Registered: 12/6/04
Re: Use of functional eqn to show that zeros of Riemann Zeta occur in
Posted: Aug 31, 2009 8:46 AM
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In article <2073967333.59072.1251644984349.JavaMail.root@gallium.mathforum.org> Luca <luca.ghislanzoni@fastwebnet.it> writes:
...
> (1) zeta(1-s) = F(s) zeta(s)
> (where with F(s) I mean to represent the part that would be too complex
> to write here without using LaTEX, and that you can instead read at eqn
> 13 of mathworld.wolfram.com/RiemannZetaFunction.html), then using it to
> prove that zeros of the Riemann Zeta function always occur in pairs.
> If everybody assumes that from (1) it must follow that if zeta(s)=0 ==>
> zeta(1-s)=0, I imagine that they are somehow confident that F(s) not= 0.


But that is false. zeta(s) = 0 for the even negative integers, but not
for any other real value, and indeed in those cases F(s) = 0. However,
for non-real s, F(s) != 0, as the only part of F(s) that could be 0
is cos(pi.s/2), which has only zeros on the real axis.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/




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