Riemann-Siegel Integral Formula
المؤلف:
Edwards, H. M.
المصدر:
"Riemann-Siegel Integral Formula" and "Alternative Proof of the Integral Formula." §7.9 and 12.6 in Riemann,s Zeta Function. New York: Dover
الجزء والصفحة:
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13-9-2019
1830
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
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