Riemann zeta function: Difference between revisions
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imported>Barry R. Smith m (Increased domain of series validity to complex nos. with imaginary part > 1.) |
imported>Barry R. Smith m (Moved complex numbers link to first appearance of "complex"/removed "positive" from sentence following Euler Product) |
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In [[mathematics]], the '''Riemann zeta function''', named after [[Bernhard Riemann]], is a [[meromorphic function]] defined for complex | In [[mathematics]], the '''Riemann zeta function''', named after [[Bernhard Riemann]], is a [[meromorphic function]] defined for [[complex number]]s with [[imaginary part]] <math>\scriptstyle \Im(s) > 1</math> by the [[infinite series]] | ||
: <math> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} </math> | : <math> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} </math> | ||
and then extended to all other | and then extended to all other complex values of ''s'' except ''s'' = 1 by [[analytic continuation]]. The function is holomorophic everywhere except for a simple pole at ''s'' = 1. | ||
[[Leonhard Euler|Euler's]] product formula for the zeta function is | [[Leonhard Euler|Euler's]] product formula for the zeta function is | ||
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: <math> \zeta(s) = \prod_{p\ \mathrm{prime}} \frac{1}{1 - p^{-s}} </math> | : <math> \zeta(s) = \prod_{p\ \mathrm{prime}} \frac{1}{1 - p^{-s}} </math> | ||
(the index ''p'' running through the whole set of | (the index ''p'' running through the whole set of [[prime number]]s). | ||
The celebrated [[Riemann hypothesis]] is the conjecture that all non-real values of ''s'' for which ζ(''s'') = 0 have real part 1/2. The problem of proving the Riemann hypothesis is the most well-known unsolved problem in mathematics. | The celebrated [[Riemann hypothesis]] is the conjecture that all non-real values of ''s'' for which ζ(''s'') = 0 have real part 1/2. The problem of proving the Riemann hypothesis is the most well-known unsolved problem in mathematics. | ||
Revision as of 17:57, 27 March 2008
In mathematics, the Riemann zeta function, named after Bernhard Riemann, is a meromorphic function defined for complex numbers with imaginary part by the infinite series
and then extended to all other complex values of s except s = 1 by analytic continuation. The function is holomorophic everywhere except for a simple pole at s = 1.
Euler's product formula for the zeta function is
(the index p running through the whole set of prime numbers).
The celebrated Riemann hypothesis is the conjecture that all non-real values of s for which ζ(s) = 0 have real part 1/2. The problem of proving the Riemann hypothesis is the most well-known unsolved problem in mathematics.