Talk:Particular values of the Riemann zeta function

Question about &zeta;(2n)
If it is defined as:


 * $$ \zeta(2n) = B_{2n} \frac{(-1)^{n+1} (2\pi)^{2n}}{2(2n)!} $$

Let's say that Pn is the nth term of the Taylor series for $$ -\frac{1}{2} \cos 2\pi $$ where the actual series is:


 * $$ \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} $$

Then:


 * $$ \zeta(2n) = B_{2n} P_n $$

Is there any significance to that, or am I going around in a circle regarding the Taylor series for ex, sin x, and cos x, and the definition of &zeta;(n)? (unsigned post by User:JVz on 15 April)


 * Yes, to both questions: there's probably significance in that, and you are going around in circles. That's a part of what makes this area of math fun: its a hall of fun-house mirrors -- its all the same, but it isn't, but it is ... linas 05:34, 2 May 2006 (UTC)


 * To add to that, for anyone who is curious, the reason you see this is because of the way it is solved: It is found by comparing the infinite product and the infinite series for sin(x)/x Minime12358 (talk) 00:04, 3 July 2013 (UTC)


 * As regards "If it is defined as:" — the value of zeta at even positive integers is not defined as the expression


 * $$ \zeta(2n) = B_{2n} \frac{(-1)^{n+1} (2\pi)^{2n}}{2(2n)!} $$.


 * That is a value that comes from a calculation.Daqu (talk) 15:03, 28 September 2015 (UTC)

This formula is not the one in the page. The formula in the page is ambiguous: should it be read (2n)! or 2(n!). In any case, it seems to me that these three versions are not coherent with the values gives after and the values of Bernoulli numbers. For example B_2=1/6 and the formula above gives (if I am not making an embarrassing mistake)


 * $$ \zeta(2)=\frac{\pi^2}{24} $$

We also have B_4=-1/30 and the formula above gives


 * $$ \zeta(2)=\frac{\pi^4}{1440} $$

So there is a mistake either in the values of even zetas, in the values of Bernoulli numbers or in the formula. I believe it's the latter. A formula that seems to work (I checked up to zeta(8)) is the one on the french wikipedia:


 * $$ \zeta(2n) = (-1)^{n+1}\frac{2^{2n-1}B_{2n}}{(2n)!}\pi^{2n} $$ — Preceding unsigned comment added by 193.50.153.33 (talk) 10:17, 3 November 2022 (UTC)


 * In my calculation rules it is the same. The page states:
 * $$\zeta(m) = (-1)^{\tfrac{m}{2}+1}\frac{(2\pi)^{m}B_m}{2 m!} \,.$$
 * And if you substitute $$m:=2n$$, you get
 * $$\zeta(2n) = (-1)^{\tfrac{2n}{2}+1}\frac{(2\pi)^{2n}B_{2n}}{2 (2n)!} = (-1)^{n+1}\frac{(2\pi)^{2n}B_{2n}}{2 (2n)!} = (-1)^{n+1}\frac{2^{2n-1}\pi^{2n}B_{2n}}{(2n)!} \,.$$
 * I can't see a mistake.–Nomen4Omen (talk) 15:41, 3 November 2022 (UTC)

Zeta derivatives
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)
 * Never mind, error appears to be elsewhere. linas 14:27, 2 May 2006 (UTC)

Definition
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)

sum of 1/(n^3)
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)

please delete this it's wrong. Thanks, Bill 99.142.20.69 (talk) 12:18, 18 April 2014 (UTC)

Zeta (x) = e
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)

Article title change
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 [//en.wikipedia.org/w/index.php?title=Zeta_constant&diff=52183904&oldid=52183213 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 reads "The values of the zeta function obtained from integral arguments are called zeta constants." However, the first editor of Zeta constant merely inserted See in [//en.wikipedia.org/w/index.php?title=Riemann_zeta_function&diff=next&oldid=19834665 his/her] edit, and later [//en.wikipedia.org/w/index.php?title=Riemann_zeta_function&diff=next&oldid=266936580 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)

getting away from Bernoulli numbers
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)

We Need examples that involve complex Numbers
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)

Utterly ridiculous use of nonstandard terminology
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)
 * OK, done. Joule36e5 (talk) 10:25, 29 November 2016 (UTC)

ζ for odd integers
Perhaps Wikipedia has some editors who are interested by new developments in open mathematical problems. $ζ$ 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)

I should note that User:David Eppstein vandalises the article for ζ(3), Apéry's constant, and possibly others. Certainly, he seems unable to make a good faith effort to improve articles. It is mystifying that someone who does not understand calculus would dispute material concerning complex analysis while rejecting cited sources that he cannot read. I believe the recommendation for resolving disputes on Wikipedia is to discuss drastic changes on talk pages. This, also, he seems incapable of. 184.155.206.191 (talk) 19:01, 10 May 2020 (UTC)


 * 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)
 * 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 &mdash; textbooks, review articles, etc. &mdash; to show that it is significant enough to be included. XOR&#39;easter (talk) 21:08, 10 May 2020 (UTC)
 * 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)

Misleading statement about "zeta(infty)" = 0
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 &middot; (Talk &middot; Contribs &middot; CentralAuth &middot; Log) 13:14, 24 July 2020 (UTC)

Prettiness or copy-ability; which is more important to you?
Calling all editors who care to comment: Which do you prefer in this context, prettiness or copy-ability? Specifically, the first of these is a prettier font and layout, and the second of these makes it easier to select the digits of the provided value (for pasting into a calculator app, etc).


 * $$\zeta^{\prime}(0) = -\frac{1}{2}\ln(2\pi)\approx -0.918938533204672741780\ldots$$



Thank you — Q uantling (talk &#124; contribs) 02:34, 4 March 2021 (UTC)
 * Comment: See discussion on my talk page. The problem here is not actually copyability; both I and Quantling are capable of selecting and copying the equation. The problem is that Wikimedia's method for displaying equations to us only allows copying the whole equation, and that makes Quantling somehow think that it is an image that is impossible to copy (somehow mysteriously unlike those other images in the form of characters from fonts that are presented when one displays text on web pages). I think that Quantling's psychological blocks are a bad reason for making mathematical formulas uglier and harder to read. Wikipedia's math formatting problems are bad enough without users running around making them worse for bad reasons. —David Eppstein (talk) 02:46, 4 March 2021 (UTC)


 * Comment: Just us, eh, @David Eppstein? I was hoping for additional opinions and insight.
 * Yes, when displayed as an equation with $$\zeta^{\prime}(0) = -\frac{1}{2}\ln(2\pi)\approx -0.918938533204672741780\ldots$$ markup a copy and paste (as text) produces the whole underlying La Te X source, which includes the number in question, and more. That is not as easy to paste into a calculator app because one wants only the digits, not the rest of the La Te X ; hence my concern about copy-ability for that case.  On the other hand, the approach with $ζ′(0) = (1/2) ln(2π) = -0.919...$ is more friendly when wants to copy and paste only the digits, but is not as pretty as the first approach.
 * I think it is safe to say that we agree that one can ultimately get the digits either way and that one can easily read and understand the text either way. There is a small copy-ability difference and a small prettiness difference.  I am hoping for additional opinions on which way to lean.  — Q uantling (talk &#124; contribs) 17:22, 6 March 2021 (UTC)
 * It is misleading to call this software LaTeX. Whoever masters this software and thinks they know LaTeX will experience a considerable shock if called upon to use actual LaTeX. Calling it TeX is closer to the truth, although of course still incorrect. Michael Hardy (talk) 18:41, 9 March 2021 (UTC)
 * Thank you for explaining that — Q uantling (talk &#124; contribs) 21:07, 9 March 2021 (UTC)


 * Comment: Maybe I'm missing the general case here, but at least for this formula couldn't one just go to the OEIS link and copy the numbers from there? Even without that, my take is readability trumps copy-pastability, after all there are likely to be much more readers than copypasters. Integral Python click here to argue with me 03:00, 7 March 2021 (UTC) EDIT: After reading the whole page, pretty much every equation has an OEIS link, except the ones with numbers provided in a table? I think that is more than sufficient for copy-ability. Integral Python click here to argue with me 03:02, 7 March 2021 (UTC)
 * Thank you @IntegralPython for your comments. The OEIS is another place that one can copy the digits from, though the format is, e.g., "3, 1, 4, 1, 5, 9" etc., with all those commas, so as with the Latex it's not ideal for pasting into a calculator.  Thank you to all who are commenting here.  — Q uantling (talk &#124; contribs) 22:45, 7 March 2021 (UTC)
 * Just been to OEIS: looks like clicking on constant after the digits gives a more user-friendly version. NeilOnWiki (talk) 16:11, 10 March 2021 (UTC)
 * Excellent! — Q uantling (talk &#124; contribs) 22:07, 10 March 2021 (UTC)


 * Comment: About copying. Since Wikipedia can be edited freely by anyone, we cannot guarantee the accuracy of the values. Also, I wouldn't like writing $$\frac{1}{2}$$ as 1/2 on pages in this area. It has multiple meanings, as in the Arithmetic operators example, and will appear on page this area at the same time.--SilverMatsu (talk) 03:48, 7 March 2021 (UTC)
 * Comment: The first option (LaTeX-style) is aesthetically preferable, and doesn't lose much functionality with regards to copy-ability, so I would agree with the others to go with the tag version. — MarkH21talk 14:09, 7 March 2021 (UTC)
 * Additional alternatives: @David Eppstein @IntegralPython @SilverMatsu @MarkH21 There are also mixed alternatives, such as the following two.  They have Latex up until but not including the actual number, thus getting much of the prettiness, and still achieving the most direct copy-ability.  If you find either of them to be better than the first two alternatives given above, please let us know.  — Q uantling (talk &#124; contribs) 22:45, 7 March 2021 (UTC)
 * $$\zeta^{\prime}(0) = -\frac{1}{2}\ln(2\pi)= $$ $-0.919$...
 * $$\zeta^{\prime}(0) = -\frac{1}{2}\ln(2\pi)= $$ $ζ′(0) = (1/2) ln(2π) = -0.919...$
 * As I have already commented when you tried doing this before, it is confusing to mix up differently styled digits in a single formula, because in most cases in mathematics when characters are differently styled there is a different meaning to them. There is also another unrelated reason for avoiding this method, coming from WP:ACCESSIBILITY: we should be using to format displayed equations, rather than :, which produces invalid html (the indentation is done by making the math be an element of a definition list, outside of any definition list for it to be an element of) and may be confusing to web-to-speech systems. However, properly formatted math displays do not allow non-math text on the same line. (For the same reason, the references to OEIS should be moved elsewhere.) —David Eppstein (talk) 23:03, 7 March 2021 (UTC)


 * Comment: My own thoughts about this have evolved a bit since first landing here from WT:WPM. I did, and still do, much prefer what you've called prettiness; which (in this context) I'd describe as improved readability and consistency of presentation. Readability helps with the aims of Wikipedia to inform; consistency will not only add to readability, but also help future editors figure out what's going on if they come to modify these equations (or consider doing so). My starting point was around the boundaries of what it means to 'inform' and I worried, and still worry, that providing artificial hooks to copy-and-paste into a calculator app crosses a line, taking Wikipedia into 'how-to' territory as a potential tool or adjunct to one. This would be okay if it came for free (as a benign side-effect), but any cost in readability and consistency seems quite a large one.
 * Having said that, the article is about Particular values of the Riemann zeta function and aiding a reader to copy-and-paste (e.g. in citing) a value fits in with its function as a list of specific values. So: how about taking your original question as a false dichotomy? Viz: ensure any values in equations are consistently formatted (ideally, to my mind, with as few decimal places as is meaningful); and perhaps add a separate table listing the full, copiable values plus OEIS links, as in the various coefficients tables or our List of mathematical constants? Hope that's useful (and palatable). NeilOnWiki (talk) 16:04, 10 March 2021 (UTC)
 * Thank you, a table could work. Let's see if others have an opinion on this.  — Q uantling (talk &#124; contribs) 22:07, 10 March 2021 (UTC)
 * What do you think of the following table? We'd have a similar table for $-0.919...$.  — Q uantling (talk &#124; contribs) 18:27, 13 March 2021 (UTC)

Table of values

 * I'm not a fan of huge numeric tables in articles. Why do we need so many of these lines? And why do we need decimal expansions for the exact rational values at the negative integers, when these values could be much more concisely written on a single line without their decimal expansions? —David Eppstein (talk) 19:04, 13 March 2021 (UTC)
 * Thank you @David Eppstein for your feedback. If you don't mind, would you also tell us what you might try instead?  For example, how many lines would you like to see, and which values?  Are you envisioning that the values that are rational would be in a separate table or some other format; or might you keep them in the same table but use decimal expansions only when a number isn't rational?  Thank you — Q uantling (talk &#124; contribs) 22:30, 13 March 2021 (UTC)
 * Thanks for taking the trouble to construct this &mdash; good to see an example to assess the pros and cons. Your own and David Eppstein's experience is probably way more than mine, but I hope this is useful feedback. My own bias is very much that "less is more", so I think we could halve the table size, either by splitting the values for positive and negative integers (i.e. two smaller tables) or by only tabulating values for $ζ′(s)$ (perhaps including $n &gt; 1$ too). As I understand it, the $0.443$-function behaves quite differently for positive and negative integers, which is reflected in the way the article is sectioned, so this would be a natural division (one table at the end of Section 2; and the other optionally in Section 3). If you did include a table with negative $-0.083$, whose values are all rational, then I'd personally prefer either repeated decimal notation (which Template:Val has an option for) or a decimal approximation to perhaps just 6 decimal places. We don't need the decimal expansion for your calculator use example (most calculators can divide), but it enables a reader to have a sense of how the values vary for varying $n = 1$, which the fractional forms tend to obscure. This table could exclude entries for $n &le; 0$, since we know from the text that these are just zero. Finally, unless there's good reason for having so many, I think it would be better to have a maximum of 18 or 20 decimal digits as in List of mathematical constants, even for $&minus;2n$, followed by the OEIS link (which could help address the concerns on copying raised above by SilverMatsu and on accessibility by David Eppstein). Good luck with it. NeilOnWiki (talk) 20:57, 15 March 2021 (UTC)

I am adding a table to the derivatives section of the article. If you don't like it you can undo it (obviously), but please, please leave a comment here indicating what you would do instead. — Q uantling (talk &#124; contribs) 00:40, 17 March 2021 (UTC)
 * I undid your change, because you formatted it in a way that made the table impossible to read in very narrow windows. In my browser (Chrome on OS X), when I make my window as narrow as possible, the table extends off the left side of the screen, making its left column invisible as it is to the left of the visible part of the window. I also get a wide horizontal scrollbar in which I can scroll rightward, but that only makes less of the table visible. Please make sure to test the accessibility of any non-standard formatting that you use before using it, preferably using multiple platforms (both browser-based and mobile), multiple browsers, and multiple window sizes. —David Eppstein (talk) 00:52, 17 March 2021 (UTC)
 * Thank you for looking out for these accessibility issues. I copied that table formatting from another Wikipedia article.  I appreciate your pointing me away from what is wrong but if you could spare the time, I would very much appreciate pointers towards what would make it better, such as to an article that does it right.  (My now knowing that one of many, many possibilities is wrong cuts down the search space a little.  Knowing which of many, many possibilities would be better cuts down the search space appreciably!) — Q uantling (talk &#124; contribs) 01:12, 17 March 2021 (UTC)
 * I don't recall trying to do a floating table recently. Maybe someone else here has more expertise on how to do that right. —David Eppstein (talk) 04:17, 17 March 2021 (UTC)

I experienced the same problem. With instinctive lack of ambition, I've placed a no-frills fully-wiki copy of the table at the end of the Derivatives section, which passes the narrowness test on Firefox. It seemed a shame to lose content which was in keeping with the article's title. I hope this version works for you both: I worry that if we need an expert every time we want a nice but inessential formatting feature that's reliably implemented we'll end up with editing becoming unnecessarily exclusive &mdash; or just not happen. What (inexpertly) seems to occur with float:right in CSS is that the right edge of the table is prohibited from crossing the right edge of the layout rectangle containing it; hence being pushed to the side as the window narrows, due largely to the wide unwrapped number entries. This also occurs for the example in Template:Float right after substituting some wide content. One solution may be to place the table in a div element with with W the table width in ems (e.g. min-width: 40em), but this is clumsy and hardly robust against future edits (other editors would benefit from an explanatory code comment to warn them). Wikibooks has an Editing Wikitext/Tables book, which may throw more light on things, though I'm unsure how far it addresses the need for accessibility amongst a broad diversity of devices and people. NeilOnWiki (talk) 13:47, 18 March 2021 (UTC)
 * What do the rest of you think about removing the "Source" column and instead have the values in the second column have a footnote that gives the OEIS source? We have a number of options for what to put in the footnote.    Also, I added a section break to this rather long talk section to break it up; I hope that is okay.  — Q uantling (talk &#124; contribs) 00:42, 19 March 2021 (UTC)


 * Thanks for initiating this discussion and following everything through I learnt a lot through it. For myself, I quite like the current Source column format with the OEIS links as they're direct, clear and simple. NeilOnWiki (talk) 17:30, 24 March 2021 (UTC)
 * Thanks for initiating this discussion and following everything through I learnt a lot through it. For myself, I quite like the current Source column format with the OEIS links as they're direct, clear and simple. NeilOnWiki (talk) 17:30, 24 March 2021 (UTC)

Truncated or rounded values
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? — Q uantling (talk &#124; contribs) 00:52, 19 March 2021 (UTC)
 * 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)