Talk:Apéry's constant

Request for expansion
This article is way too esoteric. It is a "curious number that occurs in a variety of situations", but why is it curious, and in what situations does it appear? Can someone please give a layman's description of this number? --SECurtisTX | talk 22:21, 7 March 2007 (UTC)

Merge suggestion
Since no one has expanded this article, I have changed the tag to Merge. I don't believe this is an adequate stand-alone article. It has very little meaning out of context. --SECurtisTX | talk 19:22, 20 March 2007 (UTC)


 * I disagree with the merge. There is too much information here to merge into the RZF article, and there is still more information that can be added. There are stubs out there that won't come close to the size of this article for years. darkliight[&pi;alk] 12:20, 28 March 2007 (UTC)


 * Don't merge. There is enough to justify an article on &zeta;(3). Charles Matthews 12:22, 13 May 2007 (UTC)

Random Integers
The article states that the inverse of this constant is the probability that three random integers are coprime. What distribution should these integers be drawn from? There is no such thing as a uniform distribution on all the integers. topynate 20:14, 7 August 2007 (UTC)
 * Possibly this should be in Riemann zeta function, but, it's probably indicates an attempt to encapsulate the limit as n → $$\infty$$ of the probablity that three random integers ≤ n are coprime. &mdash; Arthur Rubin | (talk) 18:55, 8 August 2007 (UTC)
 * Actual explanation is in Coprime. &mdash; Arthur Rubin |  (talk) 19:08, 8 August 2007 (UTC)

Summation
If this is correct, may I put it in the lead for compactness?

$$\zeta(3)=\sum_{k=1}^\infty\frac{1}{i^3}=1+\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+\cdots$$

Mouse is back 02:16, 18 August 2008 (UTC)
 * It's correct. I don't think anyone would mind it being added in. Cheers, Ben (talk) 08:40, 18 August 2008 (UTC)

Removal of infobox
Based upon a discussion at Wikipedia talk:WikiProject Mathematics, I've removed the infobox from the article. If anyone disagrees, could you please join the discussion there. Thanks, Paul August &#9742; 12:40, 18 October 2009 (UTC)


 * I have suggested centralizing this discussion to Wikipedia_talk:WikiProject_Mathematics and Wikipedia_talk:WikiProject_Mathematics as it refers to an infobox occurring in several articles. Please go there to build consensus on this edit. RobHar (talk) 19:34, 18 October 2009 (UTC)

Broadhurst 1998
Broadhurst 1998 isn't linking to the references the way the other are and I don't know why. Bubba73 You talkin' to me? 19:10, 3 July 2013 (UTC)

Definite Integral Expression for &zeta;(3)

 * $$\int_0^1\frac{\ln x\cdot\ln(1-x)}x\,\text{d}x\,=\,\zeta(3)$$

— 79.113.234.90 (talk) 23:54, 27 October 2013 (UTC)

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Note
In the paragraph "Fast convergence" the word "where" appears. This is bad grammar. — Preceding unsigned comment added by 81.155.192.125 (talk) 13:51, 29 September 2019 (UTC)

Attempts for finding the closed form of Apery constant
I would suggest to add the section Attempted results for the unknown closed form of Apery constant. It can mostly be found at Math StackExchange site and other websites. — Preceding unsigned comment added by QH123 (talk • contribs)
 * Hi. Math StackExchange is not what we consider a reliable source; see our guideline about web forums. If there are discussions at StackExchange that refer to peer-reviewed mathematics papers, we can use those papers. XOR&#39;easter (talk) 01:21, 24 October 2021 (UTC)

Formula shows that Apery's constant is transcendental ?
The formula $$\zeta(3) = \pi \int_0^\infty \frac{\cos(2\arctan x)}{(x^2 + 1) \left(\cosh\frac{1}{2}\pi x\right)^2} \,dx$$ at the beginning of this section of the article shows that Apery's constant is transcendental:

Set $$a = \int_0^\infty \frac{\cos(2\arctan x)}{(x^2 + 1) \left(\cosh\frac{1}{2}\pi x\right)^2} \,dx$$.

If $$a$$ is algebraic, $$\zeta(3)$$ is transcendental since the product of an algebraic and a transcendental number is transcendental. If $$a$$ is transcendental, $$\zeta(3)$$ is algebraic only if $$a$$ is equal to $$\frac{1}{\pi}$$. But if this was the case, $$\zeta(3) = 1$$. This is not the case, so $$\zeta(3)$$ is transcendental (If the formula is true). Andyloris (talk) 11:22, 22 January 2024 (UTC)


 * Incorrect. In the case that $$a$$ is transcendental, it could be any number of the form algebraic/pi and make $$\zeta(3)$$ algebraic. —David Eppstein (talk) 17:53, 22 January 2024 (UTC)

It's not clear that ζ is the Riemann zeta function
Apéry's constant does not require complex numbers. It does not refer to the Riemann Zeta function for complex numbers. Michaelmross (talk) 20:53, 29 February 2024 (UTC)


 * So? It nevertheless is the value of the zeta function at 3. —David Eppstein (talk) 21:56, 29 February 2024 (UTC)
 * I don't see the connection with the Riemann_zeta_function topic. It doesn't require analytic continuation. Michaelmross (talk) 00:58, 1 March 2024 (UTC)
 * The values of the zeta function at reals > 1 do not require analytic continuation, and yet they are still the values of the zeta function at those arguments, and still relevant to the zeta function topic. They are not relevant to the Riemann hypothesis, which is about some other values of the same function, but that does not make them irrelevant to the topic of the zeta function more generally. —David Eppstein (talk) 02:00, 1 March 2024 (UTC)