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May 31st, 2016, 09:35 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 424 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?

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: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra 
I don't think so.

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.

June 1st, 2016, 11:13 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra 
It's still finite for any particular $x$ though. The question could probably do with some clarification.

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.

June 1st, 2016, 11:37 AM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra 
Or alternatively, whether the sum approaches infinity for any $s$.


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approaching, infinity, zeta function 
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