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More on riemann zeta function

  1. Sep 19, 2008 #1
    I still don't understand a few things.

    Let's say we had a non-trivial zero counting function, [tex]Z_n(n)[/tex], for the riemann zeta function. Couldn't we fairly easily prove the riemann hypothesis by evaluating [tex]\zeta (\sigma+iZ_n)[/tex], solving for [tex] \sigma [/tex], then proving it for all n using induction?

    On another note, I still need help in evaluating the actual function. Can someone show me, step by step, how to evaluate say, [tex]\zeta (1/2 + 5i)[/tex]? Please be specific
  2. jcsd
  3. Sep 25, 2008 #2
    You can use (21) or (25) on http://mathworld.wolfram.com/RiemannZetaFunction.html in the evaluations, though you'll have to resort to numerical methods sooner or later.

    Of the attempt at proving the Riemann hypothesis, I can only say that the approach you suggest is similar to reducing the problem to Merten's conjecture (by a Mobius reciprocation)- a proof of which would imply the Riemann hypothesis! However, Merten's conjecture has been shown false (though its falsity does not imply the falsity of the Riemann hypothesis).
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