Talk:Particular values of the Riemann zeta function

Question about ζ(2n)

If it is defined as:

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

Let's say that P_n 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 e^x, sin x, and cos x, and the definition of ζ(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)

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) [1] 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 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 {{See}} 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)

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)