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Hurwitz Zeta Function

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The Hurwitz zeta function zeta(s,a) is a generalization of the Riemann zeta function zeta(s) that is also known as the generalized zeta function. It is classically defined by the formula

zeta(s,a)=sum_(k=0)^infty1/((k+a)^s)
(1)

for R[s]>1 and by analytic continuation to other s!=1, where any term with k+a=0 is excluded. It is implemented in this form in the Wolfram Language as HurwitzZeta[s, a].

The slightly different form

zeta^*(s,a)=sum_(k=0)^infty1/([(a+k)^2]^(s/2))
(2)

is implemented in the Wolfram Language as Zeta[s, a]. Note that the two are identical only for R[a]>0.

HurwitzZetaFunction

The plot above shows zeta(s,a) for real s and a, with the zero contour indicated in black.

For a>-1, a globally convergent series for zeta(s,a) (which, for fixed a, gives an analytic continuation of zeta(s,a) to the entire complex s-plane except the point s=1) is given by

zeta(s,a)=1/(s-1)sum_(n=0)^infty1/(n+1)sum_(k=0)^n(-1)^k(n; k)(a+k)^(1-s)
(3)

(Hasse 1930).

The Hurwitz zeta function is implemented in the Wolfram Language as Zeta[s, a].

For a=1, zeta(s,a) reduces to the Riemann zeta function zeta(s),

zeta(s,1)=zeta(s).
(4)

If the singular term is excluded from the sum definition of zeta(s,a), then zeta(s,0)=zeta(s) as well.

The Hurwitz zeta function is given by the integral

zeta(s,a)=1/(Gamma(s))int_0^infty(t^(s-1)dt)/(e^(at)(1-e^(-t)))
(5)

for R[s]>1 and R[a]>0.

HurwitzZetaZeros

The plot above illustrates the complex zeros of zeta(s,a) (Trott 1999), where s=x+iy. Here, the complex s-plane is horizontal and the real a-line is vertical and runs from a=1/2 at the bottom to a=1 at the top. The upper line is the critical line R[s]=1/2, which contains zeros of zeta(s)=zeta(s,1). The lower two lines are R[s]=0 and R[s]=1/2 (again), which contain zeros of 2^s-1 and zeta(s), respectively, since zeta(s,1/2)=(2^s-1)zeta(s); cf. equation (9) below.

This plot also appeared on the cover of the March 2004 issue of FOCUS, the Mathematical Association of America's news magazine.

The Hurwitz zeta function can also be given by the functional equation

zeta(s,p/q)=2Gamma(1-s)(2piq)^(s-1)sum_(n=1)^qsin((pis)/2+(2pinp)/q)zeta(1-s,n/q)
(6)

(Apostol 1995, Miller and Adamchik 1999), or the integral

zeta(s,a)=1/2a^(-s)+(a^(1-s))/(s-1)+2int_0^infty(a^2+y^2)^(-s/2){sin[stan^(-1)(y/a)]}(dy)/(e^(2piy)-1).
(7)

If R[z]<0 and 0<a<=1, then

zeta(z,a)=(2Gamma(1-z))/((2pi)^(1-z))[sin((piz)/2)sum_(n=1)^infty(cos(2pian))/(n^(1-z))+cos((piz)/2)sum_(n=1)^infty(sin(2pian))/(n^(1-z))]
(8)

(Hurwitz 1882; Whittaker and Watson 1990, pp. 268-269).

The Hurwitz zeta function satisfies

zeta(-n,a)=-(B_(n+1)(a))/(n+1)
(9)

for n>=0 (Apostol 1995, p. 264), where B_k(a) is a Bernoulli polynomial, giving the special case

zeta(0,a)=1/2-a.
(10)

In addition,

zeta(s,1/2)=sum_(k=0)^(infty)(k+1/2)^(-s)
(11)
=2^ssum_(k=0)^(infty)(2k+1)^(-s)
(12)
=2^s[zeta(s)-sum_(k=1)^(infty)(2k)^(-s)]
(13)
=2^s(1-2^(-s))zeta(s)
(14)
=(2^s-1)zeta(s).
(15)

Derivative identities include

d/(ds)zeta(0,a)=ln[Gamma(a)]-1/2ln(2pi)
(16)
d/(ds)zeta(0,0)=-1/2ln(2pi),
(17)

where Gamma(z) is the gamma function (Bailey et al. 2006, p. 179). The definition (1) implies that

d/(da)zeta(s,a)=-szeta(s+1,a)
(18)

for s!=0,1.

In the limit,

lim_(s->1)[zeta(s,a)-1/(s-1)]=-psi_0(a)
(19)

(Whittaker and Watson 1990, p. 271; Allouche 1992), where psi_0(z) is the digamma function.

The polygamma function psi_m(z) can be expressed in terms of the Hurwitz zeta function by

psi_m(z)=(-1)^(m+1)m!zeta(1+m,z).
(20)

For positive integers k, p, and q>p,

zeta^'(-2k+1,p/q)=([psi(2k)-ln(2piq)]B_(2k)(p/q))/(2k)-([psi(2k)-ln(2pi)]B_(2k))/(q^(2k)2k)+((-1)^(k+1)pi)/((2piq)^(2k))sum_(n=1)^(q-1)sin((2pipn)/q)psi_((2k-1))(n/q)+((-1)^(k+1)2(2k-1)!)/((2piq)^(2k))sum_(n=1)^(q-1)cos((2pipn)/q)zeta^'(2k,n/q)+(zeta^'(-2k+1))/(q^(2k)),
(21)

where B_n is a Bernoulli number, B_n(x) a Bernoulli polynomial, psi_n(z) is a polygamma function, and zeta(z) is the Riemann zeta function (Miller and Adamchik 1999). Miller and Adamchik (1999) also give the closed-form expressions (where a large number of typos have been corrected in the expressions below)

zeta^'(1-2k,1/2)=-(B_(2k)ln2)/(4^kk)-((2^(2k-1)-1)zeta^'(-2k+1))/(2^(2k-1))
(22)
zeta^'(1-2k,1/3; 2/3)=∓(sqrt(3)(9^k-1)B_(2k)pi)/(8k·9^k)-(3B_(2k)ln3)/(4k·9^k)∓((-1)^kpsi_(2k-1)(1/3))/(2sqrt(3)(6pi)^(2k-1))-((9^k-3)zeta^'(1-2k))/(2·9^k)
(23)
zeta^'(1-2k,1/4; 3/4)=∓((4^k-1)B_(2k)pi)/(4^(k+1)k)+((4^(k-1)-1)B_(2k)ln2)/(k·2^(4k-1))∓((-1)^kpsi_(2k-1)(1/4))/(4(8pi)^(2k-1))-((4^k-2)zeta^'(1-2k))/(2^(4k))
(24)
zeta^'(1-2k,1/6; 5/6)=∓((9^k-1)(2^(2k-1)+1)B_(2k)pi)/(8sqrt(3)k·6^(2k-1))+(B_(2k)(3^(2k-1)-1)ln2)/(4k·6^(2k-1))+(B_(2k)(2^(2k-1)-1)ln3)/(4k·6^(2k-1))∓((-1)^k(2^(2k-1)+1)psi_(2k-1)(1/3))/(2sqrt(3)(12pi)^(2k-1))+((2^(2k-1)-1)(3^(2k-1)-1)zeta^'(1-2k))/(2·6^(2k-1)),
(25)

where zeta^'(z_0,a) means dzeta(z,a)/dz|_(z=z_0), zeta^'(z_0) means dzeta(z)/dz|_(z=z_0), and the upper and lower fractions on the left side of the equations correspond to the plus and minus signs, respectively, on the right side.

See also

Hurwitz's Formula, Khinchin's Constant, Polygamma Function, QRS Constant, Riemann Zeta Function, Zeta Function

Related Wolfram sites

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Portions of this entry contributed by Jonathan Sondow (author's link)

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References

Adamchik, V. "A Class of Logarithmic Integrals." In ISSAC'97: July 21-23, 1997, Maui, Hawaii: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Ed. W. W. Kuechlin). New York: ACM, 1997.Adamchik, V. S. and Srivastava, H. M. "Some Series of the Zeta and Related Functions." Analysis 18, 131-144, 1998.Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1995.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Berndt, B. C. "On the Hurwitz Zeta-Function." Rocky Mountain J. Math. 2, 151-157, 1972.Cvijovic, D. and Klinowski, J. "Values of the Legendre Chi and Hurwitz Zeta Functions at Rational Arguments." Math. Comput. 68, 1623-1630, 1999.Elizalde, E.; Odintsov, A. D.; and Romeo, A. Zeta Regularization Techniques with Applications. River Edge, NJ: World Scientific, 1994.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Generalized Zeta Function." §1.10 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 24-27, 1981.Hasse, H. "Ein Summierungsverfahren für die Riemannsche zeta-Reihe." Math. Z. 32, 458-464, 1930.Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen für Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995.Hurwitz, A. "Einige Eigenschaften der Dirichlet'schen Funktionen F(s)=sum(D/n)·1/(n^s), die bei der Bestimmung der Klassenanzahlen Binärer quadratischer Formen auftreten." Z. für Math. und Physik 27, 86-101, 1882.Knopfmacher, J. "Generalised Euler Constants." Proc. Edinburgh Math. Soc. 21, 25-32, 1978.Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. New York: Springer-Verlag, 1966.Miller, J. and Adamchik, V. "Derivatives of the Hurwitz Zeta Function for Rational Arguments." J. Comput. Appl. Math. 100, 201-206, 1999.Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Generalized Zeta Function zeta(s,x), Bernoulli Polynomials B_n(x), Euler Polynomials E_n(x), and Polylogarithms Li_nu(x)." §1.2 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 23-24, 1990.Spanier, J. and Oldham, K. B. "The Hurwitz Function zeta(nu;u)." Ch. 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 653-664, 1987.Trott, M. "Zeros of the Generalized Riemann Zeta Function zeta(s,a) as a Function of a." Background image in graphics gallery. In Wolfram, S. The Mathematica Book, 4th ed. Cambridge, England: Cambridge University Press, p. 982, 1999. http://documents.wolfram.com/v4/MainBook/G.2.22.html.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 268-269, 1990.Wilton, J. R. "A Note on the Coefficients in the Expansion of zeta(s,x) in Powers of s-1." J. Pure Appl. Math. 50, 329-332, 1927.

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Hurwitz Zeta Function

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Sondow, Jonathan and Weisstein, Eric W. "Hurwitz Zeta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HurwitzZetaFunction.html

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