Fork me on GitHub
Math for the people, by the people.

User login

Riemann zeta function

Defines: 
Euler product formula, Riemann hypothesis
Synonym: 
$\zeta$ function
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

11M06 no label found

Comments

Dear Djao,
I feel you must add to the refrences the standard expositions/monographs on the zeta functions, e.g.
1. Titchmarsh, EC, "The Theory of the Riemann Zeta Function",
revised by DR Heath-Brown, Oxford Univ. Press, 1986.
2. Edwards, HM, "The Riemann Zeta Function",?,?...
(please find out publisher and date)
3. Karatsuba,AA and Voronin, SM (translated from Russian in English
by Neal Koblitz), "The Riemann Zeta-Function", DeGruyter Expositions in Mathematics No. 5. 1992.
4. Karatsuba AA, Complex Analysis in Number Theory, CRC Press, 1995.
5. Ivic, A
6.
7....

Regards and Best Wishes
Manoj.

I have proven RH in general case. It holds for any L-fonction.
Are you interested to read this paper? The article is written in french. My work is available in pdf format on my site http://henri.voici.org. There are two preprints (they are currently under evaluation by "Le Journal de Théorie des Nombres de Bordeaux".
I am interested in having your opinion.
Henri

Let me remind you that this article is part of the "encyclopedia" section of planetmath. An encyclopedia, by definition of encyclopedia, is not a suitable forum to use for reporting on current research. The purpose of an encyclopedia is to deliver a refined presentation of well known established facts in a manner suitable for use as a reference. It follows then that any new result, no matter what the result is, should not be reported or presented in any scientific encyclopedia until it has established itself in the refereed literature for a minimum of several years.

Your proof, whatever its merits may otherwise be, does not at this time even come close to satisfying the standards for consideration in an encyclopedia context.

I might be totally off-base here since I'm only a math-hobbyist, but it seems to me that if the zeta functional equation in this article is correct, Zeta(s) is a product of sin((pi s)/2)) and some other factors. However, as far as I know, Zeta(2) converges to a non-negative number (pi^2/6 I think??), but sin((pi 2)/2) = sin(pi) = 0

Can anyone enlighten me?

Oh nevermind. I'm sorry. This is probably the wrong place to ask questions like this since the formula is most certainly correct and I'm really asking for an explanation.

We may both be off base discussing it here but I think I can answer your question. At the positive integer s=2 where the sin factor goes to zero, the Gamma function factor is evaluated at the negative integer 1-s=-1, where it is infinite. The functional equation thus has to be interpreted as a limit at that point, and the limit is a finite nonzero number. The same thing happens for other positive even integers. But negative even integers (see the section on "Zeros of the zeta function") do behave the way you expected!

So then does that paragraph in the entry fulfill the request (in the Requests section) for the functional equation?

There is a request for a proof of the functional equation. Stating the functional equation and clarifying what it means when some of the factors become infinite does not amount to a proof of the functional equation.

> I might be totally off-base here since I'm only a
> math-hobbyist, but it seems to me that if the zeta
> functional equation in this article is correct, Zeta(s) is a
> product of sin((pi s)/2)) and some other factors. However,
> as far as I know, Zeta(2) converges to a non-negative number
> (pi^2/6 I think??), but sin((pi 2)/2) = sin(pi) =
>
> Can anyone enlighten me?

sum_{n=1}^infinity 1/n^2 = pi^2 / 6 . This can be obtained from the sum of odd ... But this IS the case if the average value of the function is zero. ... integral_0^{2pi} f(u) du = integral_0^0 f(u) du = 0 . If you think of this one term ... x_p = (4/pi) omega_n^2 (sin(t)/(omega_n^2 - 1) + sin(3t)/(omega_n^2 - 9) ...

They can be manipulated but not evaluated. Functions, on the other hand, are a little better for ... L := [[0,0],[1,3],[2,2],[6,3],[1,1]]; pointplot(L); How boring. ... I think somebody had some fun implementing plotting options in Maple. ... First, name the plots: P1 := plot([sin(t),cos(2*t+Pi/2),t=0..10*Pi]): P2 ...

Hope this Helps!

Martin Musatov

Subscribe to Comments for "Riemann zeta function"