My Math Forum A "zeta function" approaching infinity?

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 May 31st, 2016, 09:35 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 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? Thanks from manus
 June 1st, 2016, 02:37 AM #2 Senior Member   Joined: Dec 2015 From: holland Posts: 162 Thanks: 37 Math Focus: tetration I haven''t the faintest idea.
 June 1st, 2016, 03:57 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,618 Thanks: 2608 Math Focus: Mainly analysis and algebra I don't think so. Thanks from manus
 June 1st, 2016, 10:52 AM #4 Senior Member   Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT 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. Thanks from manus
 June 1st, 2016, 11:13 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,618 Thanks: 2608 Math Focus: Mainly analysis and algebra It's still finite for any particular $x$ though. The question could probably do with some clarification. Thanks from manus
 June 1st, 2016, 11:26 AM #6 Senior Member   Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT The OP asked whether the zeta function approaches infinity, not whether it ever equals infinity. Thanks from manus
 June 1st, 2016, 11:37 AM #7 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,618 Thanks: 2608 Math Focus: Mainly analysis and algebra Or alternatively, whether the sum approaches infinity for any $s$. Thanks from manus

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