The Riemann-Siegel integral formula is the following representation of the xi-function 
found in Riemann's Nachlass by Bessel-Hagen in 1926 (Siegel 1932; Edwards 2001, p. 166).
The formula is essentially
 |
(1) |
where
 |
(2) |
the symbol 
means that the path of integration is a line of slope
crossing the real axis between 0 and 1 and directed from upper left to lower right
and in which 
is defined on the slit plane (excluding 0 and negative
real numbers) by taking 
to be real on the positive real axis and setting 
(Edwards 2001, p. 167). Here, 
is analytic ar 
, 
, ..., and has a simple pole at 0.
This formula gives a proof of the functional equation
 |
(3) |
See also
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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 Resource. https://mathworld.wolfram.com/Riemann-SiegelIntegralFormula.html