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May 31st, 2016, 09:35 PM   #1
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A "zeta function" approaching infinity?

Does Z=1^(-s)+2^(-s)+3^(-s)+4^(-s)... , where s is a real or complex number of magnitude greater than one, ever approach infinity?
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June 1st, 2016, 02:37 AM   #2
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I haven''t the faintest idea.
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June 1st, 2016, 03:57 AM   #3
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I don't think so.
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June 1st, 2016, 10:52 AM   #4
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I think it does. Restrict your attention to real values of s. The zeta function is defined for all real numbers x > 1. Let x approach 1. Since $\zeta(1)$ is undefined, then $\zeta(x)$ as x approaches 1 becomes larger than any predetermined value.
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June 1st, 2016, 11:13 AM   #5
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It's still finite for any particular $x$ though. The question could probably do with some clarification.
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June 1st, 2016, 11:26 AM   #6
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The OP asked whether the zeta function approaches infinity, not whether it ever equals infinity.
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June 1st, 2016, 11:37 AM   #7
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Or alternatively, whether the sum approaches infinity for any $s$.
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