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May 7th, 2012, 05:07 AM  #1 
Senior Member Joined: Jan 2011 Posts: 120 Thanks: 2  Alternate Alternating Zeta function
I was experimenting with the following Alternate Alternating Zeta function: and believe that it has closed forms for all integers : So, for positive there seems to be a direct connection to , although these closed forms become rapidly more complex. I struggle with three questions about this function: 1) Why is with ? 2) For , the function only seems to converge when . The function appears to monotonically increase for and monotonically decrease when . Is there a logical explanation for this behavior? 3) How do I properly code in Pari/GP? I've tried the obvious: gp > ttt(s)=sumalt(n=oo,((1)^(n1))/((n+I)^s)), but that yields an error. Thanks for any help! 
May 7th, 2012, 06:10 AM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Alternate Alternating Zeta function
I'm not sure of the convergence issues here. Quote:
I get Code: ttt(s)=I^s  sumalt(n=1,(1)^n*(1/(I+n)^s+1/(In)^s))  
May 7th, 2012, 06:49 AM  #3 
Senior Member Joined: Jan 2011 Posts: 120 Thanks: 2  Re: Alternate Alternating Zeta function
That works indeed (small correction: the infinite sum should start at 0 instead of 1; I consciously added the 'i' to n, to avoid the 'div 0' in this sum). All results match with what I found using Maple and Wolfram Math online, except for the convergence of my second question. In Pari/GP the number gets smaller, where in the other tools it seems to converge to 0.061... (tested in Wolfram up to s=1/2  4000i). EDIT: It seems Pari/GP reaches a certain limit above y ~ 40ish and then the curve 'collapses'. I am particularly interested in that wave when x = 1/2 and given the results from Wolfram and Maple believe the curve should 'wave' forever around an average value. 
May 7th, 2012, 08:19 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Alternate Alternating Zeta function Quote:
Quote:
 
May 7th, 2012, 09:16 AM  #5  
Senior Member Joined: Jan 2011 Posts: 120 Thanks: 2  Re: Alternate Alternating Zeta function Quote:
Quote:
 
May 7th, 2012, 09:46 AM  #6 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Alternate Alternating Zeta function
Ah, good call. I forgot that I had pulled the  sign out front. 

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