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Riemann–Siegel formula

${\displaystyle \zeta ($s)=\sum _{n=1}^{N}{\frac {1}{n^{s}}}+\gamma (1-s)\sum _{n=1}^{M}{\frac {1}{n^{1-s}}}+R(s)}

where

${\displaystyle \gamma ($s)=\pi ^{{\tfrac {1}{2}}-s}{\frac {\Gamma \left({\tfrac {s}{2}}\right)}{\Gamma \left({\tfrac {1}{2}}(1-s)\right)}}}

is the factor appearing in the functional equation ζ(s) = γ(1 − s) ζ(1 − s), and

${\displaystyle R($s)={\frac {-\Gamma (1-s)}{2\pi i}}\int {\frac {(-x)^{s-1}e^{-Nx}}{e^{x}-1}}dx}

is a contour integral whose contour starts and ends at +∞ and circles the singularities of absolute value at most 2πM. The approximate functional equation gives an estimate for the size of the error term. Siegel (1932)^[1] and 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 (2πIm(s))^1/2. Gabcke (1979) found good bounds for the error of the Riemann–Siegel formula.

Riemann's integral formula[edit]

Riemann showed that

${\displaystyle \int _{0\searrow 1}{\frac {e^{-i\pi u^{2}+2\pi ipu}}{e^{\pi iu}-e^{-\pi iu}}}\,du={\frac {e^{i\pi p^{2}}-e^{i\pi p}}{e^{i\pi p}-e^{-i\pi p}}}}$

where the contour of integration is a line of slope −1 passing between 0 and 1 (Edwards 1974, 7.9).

He used this to give the following integral formula for the zeta function:

${\displaystyle \pi ^{-{\tfrac {s}{2}}}\Gamma \left({\tfrac {s}{2}}\right)\zeta ($s)=\pi ^{-{\tfrac {s}{2}}}\Gamma \left({\tfrac {s}{2}}\right)\int _{0\swarrow 1}{\frac {x^{-s}e^{\pi ix^{2}}}{e^{\pi ix}-e^{-\pi ix}}}\,dx+\pi ^{-{\frac {1-s}{2}}}\Gamma \left({\tfrac {1-s}{2}}\right)\int _{0\searrow 1}{\frac {x^{s-1}e^{-\pi ix^{2}}}{e^{\pi ix}-e^{-\pi ix}}}\,dx}

References[edit]

• Berry, Michael V. (1995), "The Riemann–Siegel expansion for the zeta function: high orders and remainders", Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 450 (1939): 439–462, doi:10.1098/rspa.1995.0093, ISSN 0962-8444, MR 1349513, Zbl 0842.11030
• Edwards, H.M. (1974), Riemann's zeta function, Pure and Applied Mathematics, vol. 58, New York-London: Academic Press, ISBN 0-12-232750-0, Zbl 0315.10035
• Gabcke, Wolfgang (1979), Neue Herleitung und Explizite Restabschätzung der Riemann-Siegel-Formel (in German), Georg-August-Universität Göttingen, hdl:11858/00-1735-0000-0022-6013-8, Zbl 0499.10040
• Patterson, S.J. (1988), An introduction to the theory of the Riemann zeta-function, Cambridge Studies in Advanced Mathematics, vol. 14, Cambridge: Cambridge University Press, ISBN 0-521-33535-3, Zbl 0641.10029
• Siegel, C. L. (1932), "Über Riemanns Nachlaß zur analytischen Zahlentheorie", Quellen Studien zur Geschichte der Math. Astron. Und Phys. Abt. B: Studien 2: 45–80, JFM 58.1037.07, Zbl 0004.10501 Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966.

External links[edit]

• Gourdon, X., Numerical evaluation of the Riemann Zeta-function
• Weisstein, Eric W. "Riemann–Siegel Formula". MathWorld.