Talk:Particular values of the Riemann zeta function

message: I just added the zeta derivatives at negative integers, but am not convinced these are right. By a calculation I'm doing, they seem to be off by a factor of n... Arghhhh. linas 05:34, 2 May 2006 (UTC)
replies: text: Never mind, error appears to be elsewhere. linas 14:27, 2 May 2006 (UTC)
thread_title: Zeta derivatives

message: Does the definition of the Zeta constants include the values of derivatives of the Zeta function too, or just the values of the Zeta function ? Also, by Zeta constants, do we mean only those values obtained for integer values or for complex ones too ? MP   (talk) 19:02, 8 May 2006 (UTC)
thread_title: Definition

message: the sum of 1/(n^3) is really a double-telescopic sum {1/(n^3)} for n=1 to infinity = n*[(pi^4)/90] - sum{(n-a)*[1/(a^4)]} from a=1 to n where n >> a, and n can be fixed infinitely many ways; there's NO CLOSED FORM! the answer can be found on my website... www.oddperfectnumbers.com, near the bottom of the first webpage. I'm not trying to break any guidelines, just provide an answer to this question. Enjoy! 99.135.163.205 (talk) 17:29, 25 October 2012 (UTC)
thread_title:  sum of 1/(n^3) 

message: Given their similar definitions, is there any connection between the Zeta constants and the number e ? Does the value 1.47446428731937... = Zeta-1(e)  hold any special significance ?  — Preceding unsigned comment added by 79.113.217.83 (talk) 22:36, 16 February 2013 (UTC)
thread_title:  Zeta (x) = e 

message: I changed the name of this article from Zeta constant to Particular values of Riemann zeta function, since the previous definition was dubious.
Rationale: 
1. Before, the article began as "In mathematics, a zeta constant is a value of the Riemann zeta function, with the argument being integral". But some web searches showed only pages which are probably derived from this article. In fact, the first version of this article didn't give the definition of "Zeta constant", and this edit forged the (seemingly) wrong definition. (In 2006! Sigh, it's Wikipedia after all.)
2. The current version of the page Riemann zeta function#Specific values reads "The values of the zeta function obtained from integral arguments are called zeta constants." However, the first editor of Zeta constant merely inserted  in his/her edit, and later this edit by another changed to the current one, probably just by reading Zeta constant (ugh!). 
In addition, I included a mention to non-trivial zeros, as it should be. Even if "zeta constants" were correct, the new title is more appropriate. --Teika kazura (talk) 08:02, 19 October 2013 (UTC)
thread_title:  Article title change 

message: I can define the zeta(2n) values for even "n" without using Bernoulli numbers. Would you like to see it? Bill
99.142.20.69 (talk) 12:16, 18 April 2014 (UTC)
thread_title:  getting away from Bernoulli numbers 

message: Riemann Zeta function is valid for complex numbers, so how come there are no examples of that? It's even hard to find examples while googling it. An example should be given.75.128.143.229 (talk) 02:12, 15 September 2014 (UTC)
thread_title:  We Need examples that involve complex Numbers 

message: The first and only use of the phrase "zeta constant" (or "zeta constants") in this article occurs in the section Positive integers as follows:
"The even zeta constants have the generating function:"
(which is followed by a generating function having as coefficients equal to the values of zeta at even positive integers).
It will not be clear to anyone reading the article what the phrase "zeta constant" means unless it is defined here.  
Either this phrase should be defined, or else the phrase should be replaced by its definition.  Since the phrase occurs exactly once in the article and is not standard terminology, I strongly suggest that it be replaced.Daqu (talk) 14:54, 28 September 2015 (UTC)
replies: text: OK, done.  Joule36e5 (talk) 10:25, 29 November 2016 (UTC)
thread_title:  Utterly ridiculous use of nonstandard terminology 

message: Perhaps Wikipedia has some editors who are interested by new developments in open mathematical problems.   has attractively simple representations at odd integers:
:\zeta(3) = \frac{\pi^2}{7} \lim_{\alpha \to \infty} \sum_{n=1}^\infty \frac{1}{n} \frac{\sinh(\frac{1}{\alpha}\pi n)}{\cosh(\frac{1}{\alpha}\pi n)^3}
:\zeta(5) = \frac{\pi^4}{31} \left(\frac{-1}{3}\cdot\frac{7}{\pi^2} \zeta(3) + \lim_{\alpha \to \infty} \sum_{n=1}^\infty \frac{1}{n} \frac{\sinh(\frac{1}{\alpha}\pi n)}{\cosh(\frac{1}{\alpha}\pi n)^5}\right)
:\zeta(7) = \frac{\pi^6}{127} \left[\frac{2}{45}\cdot\frac{7}{\pi^2} \zeta(3) - \frac{2}{3}\left(\frac{31}{\pi^4} \zeta(5) + \frac{1}{3}\cdot\frac{7}{\pi^2} \zeta(3)\right) + \lim_{\alpha \to \infty} \sum_{n=1}^\infty \frac{1}{n} \frac{\sinh(\frac{1}{\alpha}\pi n)}{\cosh(\frac{1}{\alpha}\pi n)^7}\right]
These expressions are simplified if T. Kyrion's limit expression is regarded as a function more elementary than ζ.  We could easily continue to list such expressions, as the development for a specific value is mainly a mechanical exercise.
184.155.206.191 (talk) 01:49, 9 May 2020 (UTC)
replies: text: For those of you playing at home this concerns attempts to add original research to the article Apéry's constant (mostly concerning numbers that are not Apéry's constant), removed twice by me and once more by . It is a mystery to me why the editor thinks it would be constructive to move the discussion here, or to frame it in the form of personal attacks. —David Eppstein (talk) 20:21, 10 May 2020 (UTC)
replies: text: It's a mystery to me, too. As to the substance of the disputed text, well, there's not a lot to say. Non-peer-reviewed arXiv preprints are not reliable sources. And even if this work were published formally, we'd need secondary or tertiary sources — textbooks, review articles, etc. — to show that it is significant enough to be included. XOR'easter (talk) 21:08, 10 May 2020 (UTC)
replies: text: We also need independent sourcing to support the claim that this is the first discovery of these recurrences. —David Eppstein (talk) 22:00, 10 May 2020 (UTC)
thread_title:  ζ for odd integers 

message: This is true but misleading to put under the even integers bit. Even if: 
\displaystyle \lim_{n \to \infty} \zeta(2n) = 1
(here, the limit is discrete ie. of the sequence a_n = \zeta(2n))
That doesn't imply that: 
\displaystyle \lim_{s \to \infty} \zeta(s)
(taken to be a continuous limit)
exists. (though clearly if it does exist, it must be equal to 1) 
A correct way to deduce this fact would be noticing that (for s > 1): 
\displaystyle \sum_{n = 1}^\infty \frac 1 {n^s} = 1 + \sum_{n = 2}^\infty \frac 1 {n^s}
and the latter sum vanishes as s \to \infty.
However I'm unsure where to put this - thoughts? --George AKA Caliburn · (Talk · Contribs · CentralAuth · Log) 13:14, 24 July 2020 (UTC)
thread_title:  Misleading statement about "zeta(infty)" = 0 

message: Some of the decimal values given in the article are truncated (as indicated by a trailing ellipsis) and others are rounded.  It seems like we should be consistent.  Which would you prefer?  —Quantling (talk | contribs) 00:52, 19 March 2021 (UTC)
replies: text: First thoughts only: I agree consistency; prefer to truncate in maths equations (less so in physical calcs); okay with either in tables, even if all equations have truncated values. NeilOnWiki (talk) 18:09, 19 March 2021 (UTC)
thread_title:  Truncated or rounded values