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The Hurwitz zeta function 
is a generalization of the Riemann zeta function 
that is also known as the generalized zeta function. It is classically defined by the formula
 | (1) |
for 
and by analytic continuation to other 
, where any term with 
is excluded. It is implemented in this form in the Wolfram Language as HurwitzZeta[s, a].
The slightly different form
 | (2) |
is implemented in the Wolfram Language as Zeta[s, a]. Note that the two are identical only for 
.
The plot above shows 
for real 
and 
, with the zero contour indicated in black.
For 
, a globally convergent series for 
(which, for fixed 
, gives an analytic continuation of 
to the entire complex 
-plane except the point 
) is given by
 | (3) |
(Hasse 1930).
The Hurwitz zeta function is implemented in the Wolfram Language as Zeta[s, a].
For 
, 
reduces to the Riemann zeta function 
,
 | (4) |
If the singular term is excluded from the sum definition of 
, then 
as well.
The Hurwitz zeta function is given by the integral
 | (5) |
for 
and 
.
The plot above illustrates the complex zeros of 
(Trott 1999), where 
. Here, the complex 
-plane is horizontal and the real 
-line is vertical and runs from 
at the bottom to 
at the top. The upper line is the critical line 
, which contains zeros of 
. The lower two lines are 
and 
(again), which contain zeros of 
and 
, respectively, since 
; cf. equation (9) below.
This plot also appeared on the cover of the March 2004 issue of FOCUS, theMathematical Association of America's news magazine.
The Hurwitz zeta function can also be given by the functional equation
 | (6) |
(Apostol 1995, Miller and Adamchik 1999), or the integral
 | (7) |
If 
and 
, then
 | (8) |
(Hurwitz 1882; Whittaker and Watson 1990, pp.268-269).
The Hurwitz zeta function satisfies
 | (9) |
for 
(Apostol 1995, p.264), where 
is a Bernoulli polynomial, giving the special case
 | (10) |
In addition,
 |  |  | (11) |
 |  |  | (12) |
 |  |  | (13) |
 |  |  | (14) |
 |  |  | (15) |
Derivative identities include
 |  |  | (16) |
 |  |  | (17) |
where 
is the gamma function (Bailey et al. 2006, p.179). The definition (1) implies that
 | (18) |
for 
.
In the limit,
 | (19) |
(Whittaker and Watson 1990, p.271; Allouche 1992), where 
is the digamma function.
The polygamma function 
can be expressed in terms of the Hurwitz zeta function by
 | (20) |
For positive integers 
, 
, and 
,
 | (21) |
where 
is a Bernoulli number, 
a Bernoulli polynomial, 
is a polygamma function, and 
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)
 |  |  | (22) |
 |  |  | (23) |
 |  |  | (24) |
 |  |  | (25) |
where 
means 
, 
means 
, 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
http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/
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 
-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 
, 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 
, Bernoulli Polynomials 
, Euler Polynomials 
, and Polylogarithms 
." §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 
." Ch.62 in An Atlas of Functions. Washington, DC: Hemisphere, pp.653-664, 1987.Trott, M. "Zeros of the Generalized Riemann Zeta Function 
as a Function of 
." 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 
in Powers of 
." J. Pure Appl. Math. 50, 329-332, 1927.
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Cite this as:
Sondow, Jonathan and Weisstein, Eric W. "Hurwitz Zeta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HurwitzZetaFunction.html
