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Talk:Riemann hypothesis

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This is an old revision of this page, as edited by FvdP (talk | contribs) at 10:54, 5 September 2002 (why zeta(-2) <> 1+4+9+...). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Can someone comment Nash's words about zeroes of Euler - Riemann zeta function ζ(s) that its zeroes are singularities of space and time. This one is of cource from Howard's film Beautiful Mind.
XJam [2002.03.24] 0 Sunday (0)

Sorry to be so dumb, but I never understood how -2, -4 etc can be zeroes of the function. Why isn't zeta(-2) just the sum of squares 1+4+9+... ? [2002.08.28] Stuart Presnell

Because obviously that would not work ! zeta(s) is defined using (sum 1/z^s) over all complex numbers s=x+iy with x > 1, then extended to the whole complex plane (excepted at -1) using analytic continuation. That is, zeta is the unique analytic function (= holomorphic function) on the complex plane (less -1) that matches the sum where it is defined. That is described in the first paragraph of the article. (There are a very few words on analytic continuation in the article Power series.) -- FvdP 10:54 Sep 5, 2002 (PDT)
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