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Riemann-Siegel Integral Formula

The Riemann-Siegel integral formula is the following representation of the xi-function xi(s) found in Riemann's Nachlass by Bessel-Hagen in 1926 (Siegel 1932; Edwards 2001, p. 166). The formula is essentially

(2xi(s))/(s(s-1))=F(s)+F(1-s^_)^_,
(1)

where

F(s)=Gamma(1/2s)pi^(-s/2)int_(0->1)(e^(-inx^2)x^(-s)dx)/(e^(ipix)-e^(-ipix)),
(2)

the symbol 0->1 means that the path of integration is a line of slope -1 crossing the real axis between 0 and 1 and directed from upper left to lower right and in which x^(-s) is defined on the slit plane (excluding 0 and negative real numbers) by taking lnx to be real on the positive real axis and setting x^(-s)=e^(-slnx) (Edwards 2001, p. 167). Here, F(s) is analytic ar -2, -4, ..., and has a simple pole at 0.

This formula gives a proof of the functional equation

xi(s)=xi(1-s).
(3)

See also

Xi-Function

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References

Edwards, H. M. "Riemann-Siegel Integral Formula" and "Alternative Proof of the Integral Formula." §7.9 and 12.6 in Riemann's Zeta Function. New York: Dover, pp. 165-170 and 273-278, 2001.Kuzmin, R. "On the Roots of Dirichlet Series." Izv. Akad. Nauk SSSR Ser. Math. Nat. Sci. 7, 1471-1491, 1934.Siegel, C. L. "Über Riemanns Nachlaß zur analytischen Zahlentheorie." Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2, 45-80, 1932. Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966.

Referenced on Wolfram|Alpha

Riemann-Siegel Integral Formula

Cite this as:

Weisstein, Eric W. "Riemann-Siegel Integral Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Riemann-SiegelIntegralFormula.html

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