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Riemann-Siegel Formula

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The Riemann-Siegel formula is a formula discovered (but not published) by Riemann for computing an asymptotic formula for the Riemann-Siegel function theta(t). The formula was subsequently discovered in an archive of Riemann's papers by C. L. Siegel (Edwards 2001, p. 136, Derbyshire 2004, pp. 257 and 263) and published by Siegel in 1932.

The Riemann-Siegel formula states that

Z(t)∼2sum_(k=1)^(nu(t))1/(sqrt(k))cos[theta(t)-tlnk]+R(t),
(1)

where

nu(t)=|_sqrt(t/(2pi))_|
(2)
R(t)=(-1)^(nu(t)-1)(t/(2pi))^(-1/4)×sum_(k=0)^(infty)c_k(sqrt(t/(2pi))-nu(t))(t/(2pi))^(-k/2)
(3)
c_k(p)=[omega^k]{exp[i(ln(t/(2pi))-1/2t-1/8pi-theta(t))]×[y^0][(sum_(j=0)^(infty)A_j(y)omega^j)(sum_(j=0)^(infty)(psi^((j))(p))/(j!)y^j)]}
(4)
A_0(y)=e^(2piiy^2)
(5)
A_j(y)=-1/2yA_(j-1)(y)-1/(32pi^2)(partial^2)/(partialy^2)(A_(j-1)(y))/y
(6)
psi(p)=(cos[2pi(p^2-p-1/(16))])/(cos(2pip))
(7)

|_x_| is the floor function (Edwards 2001), and [y^k] is coefficient notation. The first few terms c_k(p) are given by

c_0(p)=psi(p)
(8)
c_1(p)=-(psi^((3))(p))/(96pi^2)
(9)
c_2(p)=(psi^('')(p))/(64pi^2)+(psi^((6))(p))/(18432pi^4)
(10)
c_3(p)=-(psi^'(p))/(64pi^2)-(psi^((5))(p))/(3840pi^4)-(psi^((9))(p))/(5308416pi^6)
(11)
c_4(p)=(psi(p))/(128pi^2)+(19psi^((4))(p))/(24576pi^4)+(11psi^((8))(p))/(5898240pi^6)+(psi^((12))(p))/(2038431744pi^8)
(12)
c_5(p)=-(5psi^((3))(p))/(3072pi^4)-(901psi^((7))(p))/(82575360pi^6)-(7psi^((11))(p))/(849346560pi^8)-(psi^((15))(p))/(978447237120pi^(10)).
(13)

The numerators and denominators are 1, -1, 1, 1, -1, -1, -1, 1, 19, 11, 1, -5, -901, ... (OEIS A050276) and 1, 96, 64, 18432, 64, 3840, 5308416, 128, ... (OEIS A050277), respectively.

It is based on evaluation of the integral

psi(p)=(e^(ipi/8)e^(-2piip^2))/(2pii)int_Gamma(e^(iu^2/(4pi))e^(2pu))/(e^u-1)du
(14)
=(cos[2pi(p^2-p-1/(16))])/(cos(2pip)),
(15)

also denoted Psi(p), where Gamma is a line segment of slope 1, directed from upper right to lower left, which crosses the imaginary axis between 0 and 2pii (Edwards 2001, p. 147).

Another formula ascribed to Riemann and Siegel is the one presented by Riemann in his groundbreaking 1859 paper,

(pi(x)-Li(x))/((sqrt(x))/(lnx)) approx -1-2sum_(gamma in S)(sin(gammalnx))/gamma,
(16)

where pi(x) is the prime counting function, Li(x) is the logarithmic integral, and S is the set of gamma such that gamma>0 and 1/2+igamma is a (nontrivial) zero of the Riemann zeta function zeta(s). Here, the left side is the overcount of Li(x) as an estimator for the prime counting function normalized by the apparent size of the error term (Borwein and Bailey 2003, p. 68).

Alan Turing's blueprint for a zeta-function machine.

In 1939, Alan Turing designed a gear-driven analog machine intended to tabulate the cosine terms in the Riemann-Siegel sum and thereby assist a search for zeros of the Riemann zeta function. The surviving blueprint was never built and contains several mechanical errors, but it anticipates Turing's later computer calculations of zeta zeros (Casselman 2006).

See also

Logarithmic Integral, Prime Counting Function, Prime Number Theorem, Riemann-Siegel Functions, Riemann-Siegel Integral Formula, Riemann-von Mangoldt Formula, Riemann Zeta Function

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References

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 68, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Casselman, B. "Turing's Zeta-Function Machine." Not. Amer. Math. Soc. 53, 41-50, 2006. https://www.ams.org/journals/notices/200601/200601FullIssue.pdf.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Edwards, H. M. "The Riemann-Siegel Formula." Ch. 7 in Riemann's Zeta Function. New York: Dover, pp. 136-170, 2001.Granville, A. and Martin, G. "Prime Number Races." Aug. 24, 2004. https://arxiv.org/abs/math/0408319.History of Mathematics Project. "Turing's Zeta Function Machine." Image: King's College, Cambridge, Modern Archive Centre, AMT/C/2. https://www.history-of-mathematics.org/artifacts/turings-zeta-function-machine.Riemann, G. F. B. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859. Reprinted in Das Kontinuum und Andere Monographen (Ed. H. Weyl). New York: Chelsea, 1972.Sloane, N. J. A. Sequences A050276 and A050277 in "The On-Line Encyclopedia of Integer Sequences."

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Riemann-Siegel Formula

Cite this as:

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

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