Riemann-Siegel Functions

Min   Max       Re       Im

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

(1)

This function is sometimes also called the Hardy function or Hardy -function (Karatsuba and Voronin 1992, Borwein et al. 1999). The top plot superposes  (thick line) on , where  is the Riemann zeta function.

Min Max

Min   Max       Re       Im

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

			(2)
			(3)

The function  has local extrema at  (OEIS A114865 and A114866).

Values  such that

(4)

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

The series expansion of  about 0 is given by

			(5)
			(6)
			(7)

(OEIS A067626), and about  by

(8)

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

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

REFERENCES:

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 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.