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# 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)

 (8)

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

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

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

## Related Wolfram sites

http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/RiemannSiegelTheta/, http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/RiemannSiegelZ/

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## 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.

## Referenced on Wolfram|Alpha

Riemann-Siegel Functions

## Cite this as:

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