Riemann-Siegel Integral Formula

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)

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.ش