Riemann zeta function: Difference between revisions

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In [[mathematics]], the '''Riemann zeta function''', named after [[Bernhard Riemann]], is a [[meromorphic function]] defined for real numbers ''s'' > 1 by the [[infinite series]]
In [[mathematics]], the '''Riemann zeta function''', named after [[Bernhard Riemann]], is a [[meromorphic function]] defined for complex numbers 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>

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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 positive 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.