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Xi-Function


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The xi-function is the function

xi(z)=1/2z(z-1)(Gamma(1/2z))/(pi^(z/2))zeta(z)
(1)
=((z-1)Gamma(1/2z+1)zeta(z))/(sqrt(pi^z)),
(2)

where zeta(z) is the Riemann zeta function and Gamma(z) is the gamma function (Gradshteyn and Ryzhik 2000, p. 1076; Hardy 1999, p. 41; Edwards 2001, p. 16). This is a variant of the function originally defined by Riemann in his landmark paper (Riemann 1859), where the above now standard notation follows Landau (Edwards 2001, p. 16).

It is an entire function (Edwards 2001, p. 16).

It is implemented in the Wolfram Language as RiemannXi[s].

XiFunctionRoots

The zeros of xi(z) and of its derivatives are all located on the critical strip z=sigma+it, where 0<sigma<1. Therefore, the nontrivial zeros of the Riemann zeta function exactly correspond to those of xi(z) (i.e., the roots of xi(1/2+it) are the same as those of zeta(1/2+it) for real t), with the additional benefit that xi(1/2+it) is purely real.

The first few zeros occur at the values summarized in the following table (Wagon 1991, pp. 361-362 and 367-368; Havil 2003, p. 196; Odlyzko), where the corresponding negative values are also roots. The integers closest to these values are 14, 21, 25, 30, 33, 38, 41, 43, 48, 50, ... (OEIS A002410). The numbers of zeros less than 10, 10^2, 10^3, ... are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS A072080; Odlyzko).

nOEISt_n
1A05830314.134725
221.022040
325.010858
430.424876
532.935062
637.586178

Special values include

xi(0)=1/2
(3)
xi(1)=1/2
(4)
xi(2)=1/6pi
(5)
xi(3)=(3zeta(3))/(2pi)
(6)
xi(4)=1/(15)pi^2
(7)
xi(5)=(15zeta(5))/(2pi^2).
(8)

The xi function satisfies the functional equation

 xi(1-z)=xi(z)
(9)

(Edwards 2001, p. 16).

The xi-function has the Taylor series about 1/2 of

 xi(s)=sum_(n=0)^inftya_(2n)(s-1/2)^(2n),
(10)

where

 a_(2n)=4int_1^infty(d[x^(3/2)psi^'(x)])/(dx)((1/2lnx)^(2n))/((2n)!)x^(-1/4)dx
(11)

and

psi(x)=sum_(n=1)^(infty)e^(-n^2pix)
(12)
=1/2[theta_3(0,e^(-pix))-1]
(13)

(Edwards 2001, p. 15), with theta_n(z,q) a Jacobi theta function. The coefficient a_0 has the simple analytic form

a_0=-(Gamma(1/4)zeta(1/2))/(8pi^(1/4))
(14)
=0.497120778...
(15)

(OEIS A114720).

As stated by Riemann (1859) and first rigorously proved by Hadamard (1893), the xi-function can be written as

 xi(s)=xi(0)product_(rho)(1-s/rho),
(16)

where the product runs over the roots rho of xi(rho)=0 (Edwards 2001, pp. 17-21).

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The xi-function extended into the complex plane is illustrated above.

The function xi(z) is related to

 Xi(t)=xi(z),
(17)

where z=1/2+it (Gradshteyn and Ryzhik 2000, p. 1074; Edwards 2001, p. 16), which is the function originally considered and actually denoted xi(t) by Riemann (Edwards 2001, p. 16). This function can also be defined as

 Xi(it)=1/2(t^2-1/4)pi^(-t/2-1/4)Gamma(1/2t+1/4)zeta(t+1/2),
(18)

giving

 Xi(t)=-1/2(t^2+1/4)pi^(it/2-1/4)Gamma(1/4-1/2it)zeta(1/2-it).
(19)

The de Bruijn-Newman constant is defined in terms of the Xi(t) function.

Hardy (1914) proved that xi(1/2+it) has infinitely many real roots (Hardy's theorem), Hardy and Littlewood (1921) proves that the number of real roots between 0 and T is at least KT for some positive constant K and all sufficiently large T, and Selberg (1942) proved that this number is in fact at least KTlnT for some positive K and all large T (Edwards 2001, p. 19).

Coffey (2004) gives a number of formulas of derivatives of xi(s).


See also

de Bruijn-Newman Constant, Lehmer's Phenomenon, Li's Criterion, Riemann Hypothesis, Riemann-Siegel Functions, Riemann-Siegel Integral Formula, Riemann Zeta Function, Riemann Zeta Function Zeros

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References

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.Brent, R. P.; 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. II." Math. Comput. 39, 681-688, 1982.Coffey, M. W. "Relations and Positivity Results for Derivatives of the Riemann xi Function." J. Comput. Appl. Math. 166, 525-534, 2004.Conrey, J. B. "The Riemann Hypothesis." Not. Amer. Math. Soc. 50, 341-353, 2003. http://www.ams.org/notices/200303/fea-conrey-web.pdf.Edwards, H. M. "The Function xi(s)." §1.8 in Riemann's Zeta Function. New York: Dover, pp. 16-18, 2001.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, corr. enl. 4th ed. San Diego, CA: Academic Press, 2000.Hadamard, J. "Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann." J. math. pures appl. 9, 171-215, 1893.Hardy, G. H. "Sur les zéros de la fonction zeta(s) de Riemann." C. R. Acad. Sci. Paris 158, 1012-1014, 1914.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Hardy, G. H. and Littlewood, J. E. "The Zeros of Riemann's Zeta-Function on the Critical Line." Math. Z. 10, 283-317, 1921.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 202-203, 2003.Keiper, J. B. "Power Series Expansions of Riemann's xi Function." Math. Comput. 58, 765-773, 1992.Li, X.-J. "The Positivity of a Sequence of Numbers and the Riemann Hypothesis." J. Number Th. 65, 325-333, 1997.Odlyzko, A. M. "The 10^(20)th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.Odlyzko, A. "Tables of Zeros of the Riemann Zeta Function." http://www.dtc.umn.edu/~odlyzko/zeta_tables/.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. Also reprinted in English translation in Edwards, H. M. Appendix. Riemann's Zeta Function. New York: Dover, pp. 299-305, 2001.Selberg, A. "On the Zeros of Riemann's Zeta-Function." Skr. Norske Vid.-Akad. Oslo, No. 10, 1942.Sloane, N. J. A. Sequences A002410, A058303, A072080, and A114720 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.Wagon, S. "The Evidence: Where Are the Zeros of Zeta of s?" Math. Intel. 8, 57-62, 1986.Wagon, S. Mathematica in Action. New York: W. H. Freeman, 1991.

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Xi-Function

Cite this as:

Weisstein, Eric W. "Xi-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Xi-Function.html

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