Talk:Dirichlet eta function

Two relatively minor corrections

The factor 1-2^(1-s) is zero for s=1 AND also for s=1+2*k*Pi*i, k integer. So ...

1) The eta function has zeros on Re(s)=1, which do not show up very clearly on the color plot (which looks like a plot of zeta(s) instead).

2) The eta function cannot be used to define the zeta function at s=1+2*k*Pi*i, for k integer.

If you agree with this, please edit the page, and if possible provide a clearer plot.

The French Wikipedia page is already corrected: http://fr.wikipedia.org/wiki/Fonction_z%C3%AAta_de_Riemann

Jacques Gelinas —Preceding unsigned comment added by 76.67.50.57 (talk) 02:25, 13 October 2009 (UTC)

I definitely agree that the plot is not clear.

The Dirichlet eta function can still be used to define zeta(s) at those values where s = 1 + 2Kπi, K ∊ ℤ - {0}, just by taking the limit of the ratio

zeta(s) = eta(s) / (1 - 2^1-s)

as s → 1 + 2Kπi. — Preceding unsigned comment added by 2601:204:F181:9410:7461:ACCD:5996:7594 (talk) 00:44, 28 July 2024 (UTC)

Parameter in Borwein formula

Hello-

would it be possible to explain, what the "n" means in the Borwein-formula?

Thanks -

Gottfried Helms --84.138.235.250 17:07, 7 August 2006 (UTC) --Gotti 17:08, 7 August 2006 (UTC)

my guess would be that n is an arbitrary number, and the algorithm is improved in the limit as n -> infinity. Scythe33 01:14, 14 November 2006 (UTC)

General form for even positive integers

I removed a line from the bottom of the article stating:

The general form for even positive integers is:

\eta(2n) = (-1)^{n+1}{{B_{2n}\pi^{2n}(4^{n} - 1)} \over {(2n)!}}.

Carifio24 originally posted a different version of the equation. The version I have removed was a later attempt by Carifio24 to simplify the expression. However, he must have simplified incorrectly because the two equations give different results. At any rate, I don't think either of them are correct, because eta for a positive integer should always yield a positive value. If anyone knows the correct formula, please add it in! 66.243.144.178 (talk) 03:30, 22 October 2008 (UTC)

• Okay, I think I've found the correct form for even positive integers. I've added it into the article, post here if there are any problems with it. 66.243.144.178 (talk) 05:30, 22 October 2008 (UTC)

Connection to Zeta Function

Could we just enclose the proof of the relationship between the Riemann zeta function and the Dirichlet eta function? It's a very simple proof, one that wouldn't take up much space. Furthermore, when the eta function is defined in the Dirichlet series expansion, couldn't the first couple terms of the series be given? It makes it much easier to get a feel of the series. My 2 Cents' Worth (talk) 17:08, 17 July 2010 (UTC)

Suppose that Re(s)>1. Then the Dirichlet series for the zeta(s) function converges absolutely by the integral test, and we can reorder the terms at will without changing the sum. The same goes for the eta function, but only for Re(s)>1. Now, we can write for Re(s)>1, eta(s)= 1 - 2^(-s) + 3^(-s) - 4^(-s) + ... = 1 + 2^(-s) + 3^(-s) + 4^(-s) + ... - 2( 2^(-s) + 4^(-s) + ...) = zeta(s) - 2^1*2^(-s)(1 + 2^(-s) + ... ) = ( 1 - 2^(1-s) ) zeta(s). This is an equation that holds true for alls complex values of s (even s=1) by analytic continuation. Note that the factor (1 - 2^(1-s)) is zero for s=1 and also for an infinity of points on the line Re(s)=1, s= 1 + 2 i Pi k/log(2), where the equation zeta(s) = eta(s) /(1-2^(1-s)) breaks down formally (a problem often unoticed, unfortunately). 76.67.47.220 (talk) 18:32, 21 July 2010 (UTC) Jacques Gélinas

Yeah, that's the proof I'm talking about. Do we want to put it into the article, or create a link? 195.188.41.154 (talk) 09:34, 22 July 2010 (UTC)

The proof above could go in a section labelled "Relation with the zeta function", and I will do that after finding a reference for it (Euler, Titchmarsh ?). But the emphasis should remain on the eta function itself since this page is about the Dirichlet eta function, not the Riemann zeta function, avoiding too much repetition on Wikipedia. Manifestement (talk) 23:40, 15 August 2010 (UTC) Found a reference in Henrici Applied Complex Analysis book, p. 295-6, which has a very good section on Dirichlet series. Henrici writes zeta(s) = odd(s) + even(s), eta(s) = odd(s) - even(s) and then deduces the relation, pointing out the case when 2^(1-s)=1. Manifestement (talk) 07:07, 1 September 2010 (UTC)

Infinites of zeros

I don't think this is a good phrase, and if there is no objection I can rewrite the paragraph and rework sentences like "Thus the eta function has five countable infinities of zeros, while the zeta function has only three", which don't really make much sense. One problem is that if RH is true, the claim is simply false. Gene Ward Smith (talk) 00:53, 2 April 2011 (UTC). Say we use half-lines to group zeros, assuming RH. The eta function then has one infinity of zeros on the negative real axis, one infinity on the critical half-line above the real axis, a third on the critical half-line below the real axis, a fourth on the half-line {s | Re(s)=1, Im(s)>0}, and a fifth on {s | Re(s)=1, Im(s)<0}. Thanks for clearing out the ambiguity. Manifestement (talk) 19:54, 10 July 2012 (UTC)

That is an interesting and possibly useful concept to associate to an analytic function f: the smallest number of closed half-lines in the complex plane such that all zeroes of f lie in their union.

Mistake in dirichlet eta function image

I think there is a mistake in this image. Since the dirichlet eta function can be expressed in terms of the zeta function as $\eta (s)=\left(1-2^{1-s}\right)\zeta (s)$, it can't have any zeros where the zeta function doesn't. All its zeros should be in the line with real part ⁠1/2⁠. Blackbombchu (talk) 00:51, 25 March 2014 (UTC)

But $\eta$ does have zeroes where the zeta runction doesn't, because 1 - 2^1-s = 0 when s = s_K = 1 + 2πKi / ln(2), where K is any nonzero integer And, the zeta function has no pole at s_K for K ≠ 0. Hence $\eta$(s_K) = 0 for |K| ∈ ℤ+. These zeroes s_K of $\eta$ are not on the line Re(s) = 1/2, but instead on the line Re(s) = 1.

Unhelpful illustration

It helps no one to include an illustration whose colors' meaning is described only in terms of some feature of some commercial product most readers surely have never heard of.

Confusing phrase

One sentence reads as follows:

"However, in the equation η must be zero at all the points , where the denominator is zero, if the Riemann zeta function is analytic and finite there."

But to a mathematician, the word "analytic" *always* means finite. If a function has only poles at its singularities it would be called "meromorphic" (and can be viewed as a holomorphic function to the Riemann sphere ℂ ∪ {∞}).

So to add that the function is not only analytic — but also finite — immediately casts doubt over whether the writer means the same thing by "analytic" as what the reader understands that word to mean.