Riemann-Siegel Functions

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For a real positive t, the Riemann-Siegel Z function is defined by


This function is sometimes also called the Hardy function or Hardy Z-function (Karatsuba and Voronin 1992, Borwein et al. 1999). The top plot superposes Z(t) (thick line) on |zeta(1/2+it)|, where zeta(z) is the Riemann zeta function.

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For real t, the Riemann-Siegel theta function theta(t) is defined as


The function theta(t) has local extrema at (t,theta(t))=(∓6.289835...,+/-3.5309728...) (OEIS A114865 and A114866).

Values g_n such that


for n=0, 1, ... are known as Gram points (Edwards 2001, pp. 125-126).

The series expansion of theta(t) about 0 is given by


(OEIS A067626), and about infty by


(OEIS A036282 and A114721; Edwards 2001, p. 120).

These functions are implemented in the Wolfram Language as RiemannSiegelZ[z] and RiemannSiegelTheta[z].

See also

Gram's Law, Gram Point, Riemann-Siegel Formula, Riemann Zeta Function, Xi-Function

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Berry, M. V. "The Riemann-Siegel Expansion for the Zeta Function: High Orders and Remainders." Proc. Roy. Soc. London A 450, 439-462, 1995.Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Computational Strategies for the Riemann Zeta Function." J. Comput. Appl. Math. 121, 247-296, 2000.Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput. 33, 1361-1372, 1979.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Karatsuba, A. A. and Voronin, S. M. The Riemann Zeta-Function. Hawthorn, NY: de Gruyter, 1992.Odlyzko, A. M. "The 10^(20)th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.Sloane, N. J. A. Sequences A036282, A114721, A114865, and A114866 in "The On-Line Encyclopedia of Integer Sequences."Titchmarsh, E. C. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. IV." Math. Comput. 46, 667-681, 1986.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 143, 1991.

Referenced on Wolfram|Alpha

Riemann-Siegel Functions

Cite this as:

Weisstein, Eric W. "Riemann-Siegel Functions." From MathWorld--A Wolfram Web Resource.

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