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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: $\zeta_{alt}(s) := \displaystyle \sum _{n=-\infty }^{\infty }{\frac { \left( -1 \right) ^{n-1}}{\left( n+i \right) ^{s}}}$ and believe that it has closed forms for all integers $n \in \mathbb{Z}$: $\zeta_{alt}(-n)= 0$ $\zeta_{alt}(0)= -0.5+0.5 =0$ $\zeta_{alt}(1)=\pi^1 \text{csch}(\pi)$ $\zeta_{alt}(2)=\pi^2 \text{csch}(\pi) \coth(\pi)$ $\zeta_{alt}(3)=\pi^3 \left(- \frac34 i -\frac14 i \cosh(2*\pi) \right) \text{csch}(\pi)^3$ $\dots$ So, for positive $n$ there seems to be a direct connection to $\pi^n$, although these closed forms become rapidly more complex. I struggle with three questions about this function: 1) Why is $\left| \dfrac{\zeta_{alt}(y+xi)}{\zeta_{alt}(y-xi)} \right|= e^{x\pi}$ with $x,y \in \mathbb{R}$ ? 2) For $x \rightarrow \infty$, the function $\left|{\zeta_{alt}(y-xi)} \right|$ only seems to converge when $y=\frac12$. The function appears to monotonically increase for $y > \frac12$ and monotonically decrease when $y < \frac12$. Is there a logical explanation for this behavior? 3) How do I properly code ${\zeta_{alt}(s)}$ in Pari/GP? I've tried the obvious: gp > ttt(s)=sumalt(n=-oo,((-1)^(n-1))/((n+I)^s)), but that yields an error. Thanks for any help!
May 7th, 2012, 06:10 AM   #2
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Re: Alternate Alternating Zeta function

I'm not sure of the convergence issues here.

Quote:
 Originally Posted by Agno 3) How do I properly code ${\zeta_{alt}(s)}$ in Pari/GP? I've tried the obvious: gp > ttt(s)=sumalt(n=-oo,((-1)^(n-1))/((n+I)^s)), but that yields an error.
Well, writing
$\zeta_{alt}(s)= -\sum_{n=-\infty }^{\infty }\frac {(-1)^n}{(n+i)^s} = -i^{-s} - \sum_{n=1 }^{\infty }\frac {(-1)^n}{(n+i)^s}+\frac {(-1)^n}{(-n+i)^s}$
I get
Code:
ttt(s)=-I^-s - sumalt(n=1,(-1)^n*(1/(I+n)^s+1/(I-n)^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 on-line, 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 ~ 40-ish 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
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Re: Alternate Alternating Zeta function

Quote:
 Originally Posted by Agno 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
I intentionally avoided 0 so as to not double-count it. That's why I pulled it out front (i^-s).

Quote:
 Originally Posted by Agno It seems Pari/GP reaches a certain limit above y ~ 40-ish 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.
That may be a precision issue; try increasing realprecision with, e.g., \p 40

May 7th, 2012, 09:16 AM   #5
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Re: Alternate Alternating Zeta function

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by Agno 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
I intentionally avoided 0 so as to not double-count it. That's why I pulled it out front (i^-s).
Understood. However, the 'front pulled zero' should have been -(i^-s) and that indeed fixes the problem (as well, but now much more robust).

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by Agno It seems Pari/GP reaches a certain limit above y ~ 40-ish 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.
That may be a precision issue; try increasing realprecision with, e.g., \p 40
That did the trick. Even went up to 120 digits precision to plot some higher values of y, but it all works fine now. Thanks!

 May 7th, 2012, 09:46 AM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 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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