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Question about the zeta function...

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  • Jeffrey N. Cook
    All, I am a novice with the Zeta Function. Could someone help me understand something... If I set S = 1, and calculate up to 1/7 and 1/7 only, Z(s) = 1 +
    Message 1 of 2 , Jul 4, 2005
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      All,

      I am a novice with the Zeta Function. Could someone help me understand
      something...

      If I set S = 1, and calculate up to 1/7 and 1/7 only,

      Z(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + 1/7^s

      Is this what is meant "that no zeros could lie on the line Re(z) =
      1" ? Does S need to be greater than 1 here?

      Thanks,

      Jeff
    • Chris Caldwell
      ... This expression for the zeta function only converges when the real part of s is greater than one. In Calculus this was called the p-test (at least in
      Message 2 of 2 , Jul 4, 2005
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        On Mon, 4 Jul 2005, Jeffrey N. Cook wrote:
        > I am a novice with the Zeta Function. Could someone help me understand
        > something...
        >
        > If I set S = 1, and calculate up to 1/7 and 1/7 only,
        >
        > Z(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + 1/7^s

        This expression for the zeta function only converges when the real
        part of s is greater than one. In Calculus this was called the
        "p-test" (at least in the books I use).

        > Is this what is meant "that no zeros could lie on the line Re(z) =
        > 1" ? Does S need to be greater than 1 here?

        Nope. That refers to the analytic continuation of the sum above--that
        means there is a function that converges on most of the complex plane
        that agrees with the sum above (where it converges). This continuation is
        called the Riemann zeta function and its the one that all the talk is
        about

        CC
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