Talk:Riemann hypothesis

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 course from Howard's film Beautiful Mind.
XJam [2002.03.24] 0 Sunday (0)

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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 on the Riemann zeta function. See also "analytic continuation". -- FvdP Sep 5 & 7, 2002

Ah, thanks! I was relying on the little knowledge I had gleaned from a few popularisations - didn't know about the 'analytic continuation' aspect of it. If only I had bothered to read the appropriate Wikipedia entry... [2002.09.13] Stuart