Editing Riemann zeta function (section)

==Definition==
[[File:Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.pdf|thumb|upright|Bernhard Riemann's article ''On the number of primes below a given magnitude'']]

The Riemann zeta function {{math|''ζ''(''s'')}} is a function of a complex variable {{math|''s'' {{=}} ''σ'' + ''it''}}, where {{mvar|σ}} and {{mvar|t}} are real numbers. (The notation {{mvar|s}}, {{mvar|σ}}, and {{mvar|t}} is used traditionally in the study of the zeta function, following Riemann.) When {{math|1=Re(''s'') = ''σ'' > 1}}, the function can be written as a converging summation or as an integral:

<!-- This seemingly roundabout way of writing the integral makes it clear that the zeta function is a quotient of two Mellin transforms; i.e. that we integrate 1/(e^x − 1) against the invariant measure of R^* and the multiplicative character character  x^s . -->
:<math>\zeta(s) =\sum_{n=1}^\infty\frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x ^ {s-1}}{e ^ x - 1} \, \mathrm{d}x\,,</math>
where
:<math>\Gamma(s) = \int_0^\infty x^{s-1}\,e^{-x} \, \mathrm{d}x </math>
is the [[gamma function]]. The Riemann zeta function is defined for other complex values via [[analytic continuation]] of the function defined for {{math|''σ'' > 1}}.

[[Leonhard Euler]] considered the above series in 1740 for positive integer values of {{mvar|s}}, and later [[Chebyshev]] extended the definition to <math>\operatorname{Re}(s) > 1.</math><ref name='devlin'>{{cite book |last=Devlin |first=Keith |author-link=Keith Devlin |title=The Millennium Problems: The seven greatest unsolved mathematical puzzles of our time |publisher=Barnes & Noble |year=2002 |location=New York |pages=43–47 |isbn=978-0-7607-8659-8}}</ref>

The above series is a prototypical [[Dirichlet series]] that [[absolute convergence|converges absolutely]] to an [[analytic function]] for {{mvar|s}} such that {{math|''σ'' > 1}} and [[divergent series|diverges]] for all other values of {{mvar|s}}. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values {{math|''s'' ≠ 1}}. For {{math|''s'' {{=}} 1}}, the series is the [[harmonic series (mathematics)|harmonic series]] which diverges to {{math|+∞}}, and
<math display="block"> \lim_{s \to 1} (s - 1)\zeta(s) = 1.</math>
Thus the Riemann zeta function is a [[meromorphic function]] on the whole complex plane, which is [[holomorphic function|holomorphic]] everywhere except for a [[simple pole]] at {{math|''s'' {{=}} 1}} with [[Residue (complex analysis)|residue]] {{math|1}}.