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In mathematics, the '''Riemann–Siegel formula''' is an asymptotic formula for the error of the '''approximate functional equation''' of the [[Riemann zeta function]], an approximation of the zeta function by a sum of two finite [[Dirichlet series]]. It was found by {{harvtxt|Siegel|1932}} in unpublished manuscripts of [[Bernhard Riemann]] dating from the 1850s. Siegel derived it from the '''Riemann–Siegel integral formula''', an expression for the zeta function involving [[contour integral]]s. It is often used to compute values of the Riemann–Siegel formula, sometimes in combination with the [[Odlyzko–Schönhage algorithm]] which speeds it up considerably. When used along the critical line, it is often useful to use it in a form where it becomes a formula for the [[Z function]].
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If ''M'' and ''N'' are non-negative integers, then the zeta function is equal to
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:<math>\zeta(s) = \sum_{n=1}^N\frac{1}{n^s} + \gamma(1-s)\sum_{n=1}^M\frac{1}{n^{1-s}} + R(s) </math>
 
where
 
:<math>\displaystyle\gamma(s) = \pi^{1/2-s}\Gamma(s/2)/\Gamma((1-s)/2)</math>
 
is the factor appearing in the functional equation ζ(''s'')&nbsp;=&nbsp;γ(''s'')&nbsp;ζ(1&nbsp;&minus;&nbsp;''s''), and
 
:<math>R(s) = \frac{-\Gamma(1-s)}{2\pi i}\int \frac{(-x)^{s-1}e^{-Nx}dx}{e^x-1}</math>
 
is a contour integral whose contour starts and ends at +∞ and circles the singularities of absolute value at most 2π''M''. The approximate functional equation gives an estimate for the size of the error term.
{{harvtxt|Siegel|1932}} and {{harvtxt|Edwards|1974}} derive the Riemann–Siegel formula from this by applying the [[method of steepest descent]] to this integral to give an asymptotic expansion for the error term ''R''(''s'') as a series of negative powers of Im(''s'').  
In applications ''s'' is usually on the critical line, and the positive integers ''M'' and ''N'' are chosen to be about (2π&nbsp;Im(''s''))<sup>1/2</sup>. {{harvtxt|Gabcke|1979}} found good bounds for the error of the Riemann–Siegel formula.
 
==Riemann's integral formula==
Riemann showed that
 
:<math>\int_{0\searrow 1}\frac{e^{-i\pi u^2+2\pi i pu}du}{e^{\pi i u}-e^{-\pi i u}} =
\frac{e^{i\pi p^2}-e^{i\pi p}}{e^{i\pi p}- e^{-i\pi p}}</math>
 
where the contour of integration is a line of slope &minus;1 passing between 0 and 1 {{harv|Edwards|1974|loc=7.9}}.
 
He used this to give the following integral formula for the zeta function:
 
:<math>\displaystyle\pi^{-s/2}\Gamma(s/2)\zeta(s)=</math>
 
:<math> \pi^{-s/2}\Gamma(s/2)\int_{0\swarrow 1}\frac{x^{-s}e^{\pi i x^2}}{e^{\pi i x}-e^{-\pi i x}}\,dx
+\pi^{-(1-s)/2}\Gamma((1-s)/2)\int_{0\searrow 1}\frac{x^{s-1}e^{-\pi i x^2}}{e^{\pi i x}-e^{-\pi i x}}\,dx
</math>
 
==References==
*{{Citation | last1=Berry | first1=Michael V. | author1-link=Michael Berry (physicist) | title=The Riemann–Siegel expansion for the zeta function: high orders and remainders | doi=10.1098/rspa.1995.0093 | mr=1349513 | year=1995 | journal=Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences | issn=0962-8444 | volume=450 | issue=1939 | pages=439–462 | zbl=0842.11030 }}
*{{citation | last=Edwards | first=H.M. | authorlink=Harold Edwards (mathematician) | title=Riemann's zeta function | series=Pure and Applied Mathematics | volume=58 | location=New York-London |publisher=Academic Press | year=1974 | isbn=0-12-232750-0 | zbl=0315.10035}}
*{{citation | first=Wolfgang | last=Gabcke | year=1979 | publisher=Georg-August-Universität Göttingen | url=http://www.worldcat.org/oclc/60563505|title=Neue Herleitung und Explizite Restabschätzung der Riemann-Siegel-Formel | zbl=0499.10040 | language=German }}
* {{citation | last=Patterson | first=S.J. | title=An introduction to the theory of the Riemann zeta-function | series=Cambridge Studies in Advanced Mathematics | volume=14 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1988 | isbn=0-521-33535-3 | zbl=0641.10029 }}
*{{citation | authorlink=Carl Ludwig Siegel | last=Siegel | first=C. L. | title=Über Riemanns Nachlaß zur analytischen Zahlentheorie | journal= Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2 | pages= 45–80 | year= 1932 | zbl=0004.10501 | jfm=58.1037.07 }} Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: [[Springer-Verlag]], 1966.
 
==External links==
*{{citation|first=X.|last=Gourdon|url=http://numbers.computation.free.fr/Constants/Miscellaneous/zetaevaluations.html|title=Numerical evaluation of the Riemann Zeta-function}}
*{{mathworld|urlname=Riemann-SiegelFormula|title=Riemann–Siegel Formula}}
 
{{DEFAULTSORT:Riemann-Siegel Formula}}
[[Category:Zeta and L-functions]]
[[Category:Theorems in number theory]]

Latest revision as of 23:13, 23 May 2014

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