My Math Forum Question on the Dirichlet eta function

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 April 5th, 2012, 10:48 PM #1 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 Question on the Dirichlet eta function OK, I am not sure if I should post this in number theory, but since it is related to Zeta... So the Dirichlet eta function is $\sum_n\frac{(-1)^n}{n^s}$. Now it is known it is convergent for Re(s)>0. My questions: 1) I always assumed that was true. Today I tried to derive why it is convergent, but I could not (actually I can of course if s is real using alternate series criteria, but not if s is complex since it is not monotone anymore...) Does somebody know and could give me a hint please? Like which theorem to use? I think it is related to abel's work, but I could not find a theorem that suites this series 2) Do we know how fast should this series converge? I mean I found computationally what seems to be the formula for the envelope of convergence. Is there a theorem stating how fast it converges? Thanks!
 April 5th, 2012, 11:16 PM #2 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: Question on the Dirichlet eta function This is definitely a number theory question. I'm sick today and not quite thinking straight, so I don't think I'll be of any help on #1. It converges extremely slowly. Series acceleration is needed for reasonable computations. For |s-1|, |Re s - 1/2|, and |Im s| large I would just use the functional equatin relating it to zeta. (Otherwise you may have trouble with zeta zeros, but either you can compute it with some care through that method or you can revert to another standard technique.)
 April 5th, 2012, 11:40 PM #3 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 Re: Question on the Dirichlet eta function OK, no problem for #1, even in perfect health now, I could not figure it out today anyway! If you have an idea another time... I don't know how to post a pdf file here, but I found a fitting for the convergence of for instance $\sum^N_n\frac{(-1)^n\sin(t\log(n))}{n^{\sigma}$ and I don't agree with what you are saying, it converges not slowly at all (but perhaps if Re(s) is very very small). Actually I found (and it matches so well...) that it converges as $\eta(s)+\frac{1}{2N^{\sigma}}$ with . I am sure there also exists a theorem on this, but again I don't know where to look for, especially I first need to understand why it converges before understanding how it converges...thanks anyways..
April 5th, 2012, 11:59 PM   #4
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Re: Question on the Dirichlet eta function

Quote:
 Originally Posted by Dougy it converges not slowly at all (but perhaps if Re(s) is very very small).
I only get 11 digits of precision when I compute the sum with a million terms (about 1 second) at s = 2. That's pretty slow by my standards, and Re(s) is surely not that small. Using 100 million terms only increases the number of correct digits to 15, and that makes the calculation time balloon to 2 minutes.

By comparison, it takes only 15 microseconds to find 1000 digits using the zeta relation.

So either we have very different standards (for "slow" or "small") or else we're talking about different things. The latter is of course always possible, and especially so right now...

 April 6th, 2012, 12:11 AM #5 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 Re: Question on the Dirichlet eta function I think we are taking about the same thing, I wrote above the series until N. But,yeah I was only converging to five digits, but instead of taking about digits, let's put it mathematically... it converges as $\frac{1}{2N^{\sigma}}$ (at least my computer tell me so, I can not derive this). So for instance for the famous Re(s)=0.5 it converges as $\frac{1}{2\sqrt{N}}$, I don't know what are you standard but I don't consider this a small convergence, everything is relative I guess!
 April 6th, 2012, 12:38 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: Question on the Dirichlet eta function Extrapolating from my results it would take 10^500 terms and over 10^490 years to get a thousand digits. The competing method takes millionths of a second to get the same precision. If I use your computers' result it's more like 10^240 years, but either way it's infeasible to get precise results through direct calculation through the definition.
 April 6th, 2012, 12:51 AM #7 Senior Member   Joined: Nov 2011 Posts: 595 Thanks: 16 Re: Question on the Dirichlet eta function Actually I found a "pseudo" derivation of this, it's late so I will do fast.We need to calculate $S_N=\sum^N_n\frac{(-1)^n}{n^s}$. Now let us regroup terms by two (that's where I am not sure if it is allowed, I think yes from Abel'sumation, any order should give the same result..) so we call $v_n=\frac{1}{n^s}-\frac{1}{(n+1)^s}$ equals $\frac{1}{n^s}(1-(1+\frac{1}{n})^{-s})$ for large n, this becomes $v_n=\frac{s}{n^{s+1}}$ so that $S_N=\sum^N_n\frac{s}{n^{s+1}}\$ and of course we conclude that $S_N\~\frac{1}{N^{s}}\$. now computationally, I found that $S_N\~\frac{1}{2N^{s}}\$. Now it fits so perfectly, for different s, that I doubt the latter is wrong so perhaps something is missing in what I wrote but I think the glance of the idea is there and at the same time why it converges. Still, I wish to find a formal derivation somewhere but I will content myself with this for now!
April 6th, 2012, 06:27 AM   #8
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Re: Question on the Dirichlet eta function

Quote:
 Originally Posted by Dougy Now let us regroup terms by two (that's where I am not sure if it is allowed, I think yes from Abel'sumation, any order should give the same result..)
Well it's certainly fine when the sum converges absolutely, so if |s| > 1 you're fine. It may work elsewhere as well.

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