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Riemann zeta function (Definition)

"Riemann zeta function" is owned by alozano. [ full author list (2) | owner history (1) ]
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See Also: analytic continuation of Riemann zeta to critical strip, Dedekind zeta function, Dirichlet series, Euler product, complex, Euler product formula

Other names:  $\zeta$ function
Also defines:  Euler product formula, Riemann hypothesis

Attachments:
Riemann $þeta$-function (Definition) by PrimeFan
Riemann $\Xi$ function (Definition) by PrimeFan
Riemann $\varpi$ function (Definition) by rspuzio
Apéry's constant (Definition) by bbukh
formulae for zeta in the critical strip (Theorem) by mathcam
functional equation of the Riemann zeta function (Definition) by mathcam
value of the Riemann zeta function at $s=2$ (Theorem) by alozano
values of the Riemann zeta function in terms of Bernoulli numbers (Theorem) by Mathprof
value of the Riemann zeta function at $s=0$ (Theorem) by Wkbj79
critical strip (Definition) by Wkbj79
convergence of Riemann zeta series (Definition) by pahio
Riemann zeta function has no zeros on $\Re s=0,1$ (Theorem) by rm50
value of Riemann zeta function at $s = 4$ (Example) by pahio
Euler product formula (Theorem) by pahio
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Cross-references: consequences, hypothesis, equation, equivalent, critical strip, NOR, region, real part, root, proofs, addition, arithmetic progressions, Dirichlet's theorem on primes in arithmetic progressions, prime number theorem, lines, analysis, pole, limit, formula, finite, distribution, analytic, link, gamma function, functional equation, residue, simple pole, complex plane, entire, meromorphic continuation, neighborhood, converges uniformly, primes, integer, positive, product, properties, key, complex numbers, absolutely convergent, valid, series, function, complex
There are 59 references to this entry.

This is version 15 of Riemann zeta function, born on 2002-05-06, modified 2007-05-18.
Object id is 2896, canonical name is RiemannZetaFunction.
Accessed 43369 times total.

Classification:
AMS MSC11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)

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Is the functional equation correct? by byungsoolee on 2005-10-23 18:20:48
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?
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Proof by henri on 2004-07-31 01:00:03

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
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Addition to References... by Manoj on 2003-07-17 17:08:24
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.
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