Editing Hurwitz zeta function

{{short description|Special function in mathematics}}

In [[mathematics]], the '''Hurwitz zeta function''' is one of the many [[zeta function]]s. It is formally defined for [[complex number|complex]] variables {{mvar|s}} with {{math|Re(''s'') > 1}} and {{math|''a'' ≠ 0, −1, −2, …}} by

:<math>\zeta(s,a) = \sum_{n=0}^\infty \frac{1}{(n+a)^{s}}.</math>

This series is [[absolutely convergent]] for the given values of {{mvar|s}} and {{mvar|a}} and can be extended to a [[meromorphic function]] defined for all {{math|''s'' ≠ 1}}. The [[Riemann zeta function]] is {{math|&zeta;(''s'',1)}}. The Hurwitz zeta function is named after [[Adolf Hurwitz]], who introduced it in 1882.<ref>{{cite journal |first=Adolf |last=Hurwitz |author-link=Adolf Hurwitz |title=Einige Eigenschaften der Dirichlet'schen Functionen <math display="inline">F(s) = \sum \left(\frac{D}{n}\right) \cdot \frac{1}{n}</math>, die bei der Bestimmung der Classenanzahlen binärer quadratischer Formen auftreten |journal=Zeitschrift für Mathematik und Physik |volume=27 |pages=86–101 |lang=de |date=1882 |url=https://archive.org/details/zeitschriftfurm13unkngoog/page/n95}}</ref>

[[File:Hurwitza1ov3v2.png|right|thumb|Hurwitz zeta function corresponding to {{math|1=''a'' = 1/3}}.  It is generated as a [[Matplotlib]] plot using a version of the [[Domain coloring]] method.<ref>{{Cite web|url=http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb|title=Jupyter Notebook Viewer}}</ref>]]
[[File:Hurwitza24ov25v2.png|right|thumb|Hurwitz zeta function corresponding to {{math|1=''a'' = 24/25}}.]] 
[[File:HurwitzofAz3p4j.png|right|thumb| Hurwitz zeta function as a function of {{mvar|a}} with {{math|1=''s'' = 3 + 4''i''}}.]]

==Integral representation==
The Hurwitz zeta function has an integral representation
:<math>\zeta(s,a) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}e^{-ax}}{1-e^{-x}} dx</math>
for <math>\operatorname{Re}(s)>1</math> and <math>\operatorname{Re}(a)>0.</math> (This integral can be viewed as a [[Mellin transform]].) The formula can be obtained, roughly, by writing
:<math>\zeta(s,a)\Gamma(s)
= \sum_{n=0}^\infty \frac{1}{(n+a)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x}
= \sum_{n=0}^\infty \int_0^\infty y^s e^{-(n+a)y} \frac{dy}{y}</math>
and then interchanging the sum and integral.<ref>{{harvnb|Apostol|1976|p=251|loc=Theorem 12.2}}</ref>

The integral representation above can be converted to a [[contour integral]] representation
:<math>\zeta(s,a) = -\Gamma(1-s)\frac{1}{2 \pi i} \int_C \frac{(-z)^{s-1}e^{-az}}{1-e^{-z}} dz</math>
where <math>C</math> is a [[Hankel contour]] counterclockwise around the positive real axis, and the [[principal branch]] is used for the [[complex exponentiation]] <math>(-z)^{s-1}</math>. Unlike the previous integral, this integral is valid for all ''s'', and indeed is an [[entire function]] of ''s''.<ref>{{harvnb|Whittaker|Watson|1927|p=266|loc=Section 13.13}}</ref>

The contour integral representation provides an [[analytic continuation]] of <math>\zeta(s,a)</math> to all <math>s \ne 1</math>. At <math>s = 1</math>, it has a [[simple pole]] with [[residue (complex analysis)|residue]] <math>1</math>.<ref>{{harvnb|Apostol|1976|p=255|loc=Theorem 12.4}}</ref>

==Hurwitz's formula==
The Hurwitz zeta function satisfies an identity which generalizes the [[Riemann zeta function#Riemann's functional equation|functional equation of the Riemann zeta function]]:<ref name="apostol-theorem-12-6">{{harvnb|Apostol|1976|p=257|loc=Theorem 12.6}}</ref>
:<math>\zeta(1-s,a) = \frac{\Gamma(s)}{(2\pi)^s} \left( e^{-\pi i s/2} \sum_{n=1}^\infty \frac{e^{2\pi ina}}{n^s} + e^{\pi i s/2} \sum_{n=1}^\infty \frac{e^{-2\pi ina}}{n^s} \right),</math>
valid for Re(''s'') > 1 and 0 < ''a'' ≤ 1. The Riemann zeta functional equation is the special case ''a'' = 1:<ref>{{harvnb|Apostol|1976|p=259|loc=Theorem 12.7}}</ref>
:<math>\zeta(1-s) = \frac{2\Gamma(s)}{(2\pi)^{s}} \cos\left(\frac{\pi s}{2}\right) \zeta(s)</math>

Hurwitz's formula can also be expressed as<ref name="whittaker-watson-section-13-15">{{harvnb|Whittaker|Watson|1927|pp=268–269|loc=Section 13.15}}</ref>
:<math>\zeta(s,a) = \frac{2\Gamma(1-s)}{(2\pi)^{1-s}} \left( \sin\left(\frac{\pi s}{2}\right) \sum_{n=1}^\infty \frac{\cos(2\pi na)}{n^{1-s}} + \cos\left(\frac{\pi s}{2}\right) \sum_{n=1}^\infty \frac{\sin(2\pi na)}{n^{1-s}} \right)</math>
(for Re(''s'') < 0 and 0 < ''a'' ≤ 1).

Hurwitz's formula has a variety of different proofs.<ref>See the references in Section 4 of: {{cite journal |first1= S. |last1= Kanemitsu |first2= Y. |last2= Tanigawa |first3= H. |last3= Tsukada |first4= M. |last4= Yoshimoto |title= Contributions to the theory of the Hurwitz zeta-function |journal= [[Hardy-Ramanujan Journal]] |volume= 30| date= 2007 |pages= 31–55 |doi= 10.46298/hrj.2007.159 |zbl= 1157.11036|doi-access= free }}</ref> One proof uses the contour integration representation along with the [[residue theorem]].<ref name="apostol-theorem-12-6" /><ref name="whittaker-watson-section-13-15" /> A second proof uses a [[theta function]] identity, or equivalently [[Poisson summation]].<ref>{{cite journal |first=N. J. |last=Fine |author-link= Nathan Fine |title= Note on the Hurwitz Zeta-Function |journal= [[Proceedings of the American Mathematical Society]] |volume= 2 |number= 3 |date= June 1951 |pages= 361–364 |doi= 10.2307/2031757 |jstor=2031757 |doi-access=free |zbl= 0043.07802}}</ref> These proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in [[On the Number of Primes Less Than a Given Magnitude|Riemann's 1859 paper]]. Another proof of the Hurwitz formula uses [[Euler–Maclaurin summation]] to express the Hurwitz zeta function as an integral
:<math>\zeta(s,a) = s \int_{-a}^\infty \frac{\lfloor x \rfloor - x + \frac{1}{2}}{(x+a)^{s+1}} dx</math>
(−1 < Re(''s'') < 0 and 0 < ''a'' ≤ 1) and then expanding the numerator as a [[Fourier series]].<ref>{{cite journal |first= Bruce C. |last= Berndt |author-link= Bruce C. Berndt |title= On the Hurwitz zeta-function |journal= Rocky Mountain Journal of Mathematics |volume= 2 |number= 1 |date= Winter 1972 |pages= 151–158 |doi= 10.1216/RMJ-1972-2-1-151 |zbl= 0229.10023|doi-access= free }}</ref>

===Functional equation for rational ''a''===
When ''a'' is a rational number, Hurwitz's formula leads to the following [[functional equation]]: For integers <math>1\leq m \leq n </math>,
:<math>\zeta \left(1-s,\frac{m}{n} \right) =
\frac{2\Gamma(s)}{ (2\pi n)^s }
\sum_{k=1}^n \left[\cos
\left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\;
\zeta \left( s,\frac {k}{n} \right)\right]
</math>
holds for all values of ''s''.<ref>{{harvnb|Apostol|1976|p=261|loc=Theorem 12.8}}</ref>

This functional equation can be written as another equivalent form:

<math> \zeta \left(1-s,\frac{m}{n} \right) = \frac{\Gamma(s)}{ (2\pi n)^s} \sum_{k=1}^n \left[e^{\frac{\pi is}{2}}e^{-\frac{2\pi ikm}{n}}\zeta \left( s,\frac {k}{n} \right) + e^{-\frac{\pi is}{2}}e^{\frac{2\pi ikm}{n}}\zeta \left( s,\frac {k}{n} \right) \right]
</math>.

==Some finite sums==
Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form
:<math>
\sum_{r=1}^{m-1} \zeta\left(s,\frac{r}{m}\right) 
\cos\dfrac{2\pi rk}{m} =\frac{m \Gamma(1-s)}{(2\pi m)^{1-s}}
\sin\frac{\pi s}{2} \cdot \left\{\zeta\left(1-s,\frac{k}{m}\right) +
\zeta\left(1-s,1-\frac{k}{m}\right) \right\}  - \zeta(s) 
</math>

:<math>
\sum_{r=1}^{m-1} \zeta\left(s,\frac{r}{m}\right) 
\sin\dfrac{2\pi rk}{m}= \frac{m \Gamma(1-s)}{(2\pi m)^{1-s}}
\cos \frac{\pi s}{2} \cdot \left\{\zeta\left(1-s,\frac{k}{m}\right) -
\zeta\left(1-s,1-\frac{k}{m}\right)\right\}  
</math>

:<math>
\sum_{r=1}^{m-1} \zeta^2\left(s,\frac{r}{m}\right) = 
\big(m^{2s-1}-1 \big)\zeta^2(s) + \frac{2m\Gamma^2(1-s)}{(2\pi m)^{2-2s}}
\sum_{l=1}^{m-1} \left\{\zeta\left(1-s,\frac{l}{m}\right) - \cos\pi s 
\cdot \zeta\left(1-s,1-\frac{l}{m}\right)\right\} \zeta\left(1-s,\frac{l}{m}\right)
</math>
where ''m'' is positive integer greater than 2 and ''s'' is complex, see e.g. Appendix B in.<ref>{{cite journal|doi=10.1016/j.jnt.2014.08.009 |first=I.V. |last=Blagouchine |title=A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations |journal=Journal of Number Theory |publisher=Elsevier |volume=148 |pages=537–592 |date=2014 |arxiv=1401.3724}}</ref>

==Series representation==
A convergent [[Newton series]] representation defined for (real) ''a'' > 0 and any complex ''s'' &ne; 1 was given by [[Helmut Hasse]] in 1930:<ref>{{Citation |first=Helmut |last=Hasse |title=Ein Summierungsverfahren für die Riemannsche ζ-Reihe |year=1930 |journal=[[Mathematische Zeitschrift]] |volume=32 |issue=1 |pages=458–464 |doi=10.1007/BF01194645 | jfm=56.0894.03 |s2cid=120392534 | url=https://eudml.org/doc/168238 }}</ref>

:<math>\zeta(s,a)=\frac{1}{s-1}
\sum_{n=0}^\infty \frac{1}{n+1}
\sum_{k=0}^n (-1)^k {n \choose k} (a+k)^{1-s}.</math>

This series converges uniformly on [[compact subset]]s of the ''s''-plane to an [[entire function]]. The inner sum may be understood to be the ''n''th [[forward difference]] of <math>a^{1-s}</math>; that is,

:<math>\Delta^n a^{1-s} = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (a+k)^{1-s}</math>

where &Delta; is the [[forward difference operator]]. Thus, one may write:

:<math>\begin{align}
  \zeta(s, a) &= \frac{1}{s-1}\sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n a^{1-s}\\
              &= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} a^{1-s}
\end{align}</math>

==Taylor series==
The partial derivative of the zeta in the second argument is a [[sheffer sequence|shift]]:

:<math>\frac {\partial} {\partial a} \zeta (s,a) = -s\zeta(s+1,a).</math>

Thus, the [[Taylor series]] can be written as:

:<math>\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!}
\frac {\partial^k} {\partial x^k} \zeta (s,x) =
\sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).</math>

Alternatively,

:<math>\zeta(s, q) = \frac{1}{q^s} + \sum_{n=0}^{\infty} (-q)^n {s + n - 1 \choose n} \zeta(s + n),</math>

with <math>|q| < 1</math>.<ref>{{cite journal |last=Vepstas |first=Linas |arxiv=math/0702243 |title=An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions |year=2007 |doi=10.1007/s11075-007-9153-8 |volume=47 |issue=3 |journal=Numerical Algorithms |pages=211–252|bibcode=2008NuAlg..47..211V|s2cid=15131811 }}</ref>

Closely related is the '''Stark–Keiper''' formula:

:<math>\zeta(s,N) =
\sum_{k=0}^\infty \left[ N+\frac {s-1}{k+1}\right]
{s+k-1 \choose s-1} (-1)^k \zeta (s+k,N) </math>

which holds for integer ''N'' and arbitrary ''s''. See also [[Faulhaber's formula]] for a similar relation on finite sums of powers of integers.

==Laurent series==
The [[Laurent series]] expansion can be used to define generalized [[Stieltjes constants]] that occur in the series
:<math>\zeta(s,a) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(a) (s-1)^n.</math>

In particular, the constant term is given by
:<math>\lim_{s\to 1} \left[ \zeta(s,a) - \frac{1}{s-1}\right] =
\gamma_0(a)=
\frac{-\Gamma'(a)}{\Gamma(a)} = -\psi(a)</math>
where <math>\Gamma</math> is the [[gamma function]] and <math>\psi = \Gamma' / \Gamma</math> is the [[digamma function]]. As a special case, <math>\gamma_0(1) = -\psi(1) = \gamma_0 = \gamma</math>.

==Discrete Fourier transform==
The [[discrete Fourier transform]] of the Hurwitz zeta function with respect to the order ''s'' is the [[Legendre chi function]].<ref>{{cite journal |last=Jacek Klinowski |first=Djurdje Cvijović | title=Values of the Legendre chi and Hurwitz zeta functions at rational arguments |journal=[[Mathematics of Computation]] |volume=68 |date=1999|issue=228 |pages=1623–1631 |doi=10.1090/S0025-5718-99-01091-1|bibcode=1999MaCom..68.1623C |doi-access=free }}</ref>

==Particular values==

===Negative integers===
The values of ''ζ''(''s'', ''a'') at ''s'' = 0, −1, −2, ... are related to the [[Bernoulli polynomials]]:<ref>{{harvnb|Apostol|1976|p=264|loc=Theorem 12.13}}</ref>
:<math>\zeta(-n,a) = -\frac{B_{n+1}(a)}{n+1}.</math>
For example, the <math>n=0</math> case gives<ref>{{harvnb|Apostol|1976|p=268}}</ref>
:<math>\zeta(0,a) = \frac{1}{2} - a.</math>

===''s''-derivative===
The [[partial derivative]] with respect to ''s'' at ''s'' = 0 is related to the gamma function:
:<math>\left. \frac{\partial}{\partial s} \zeta(s,a) \right|_{s=0} = \log\Gamma(a) - \frac{1}{2} \log(2\pi)</math>
In particular, <math display="inline">\zeta'(0) = -\frac{1}{2} \log(2\pi).</math> The formula is due to [[Mathias Lerch|Lerch]].<ref>{{cite journal |last=Berndt |first=Bruce C. |author-link=Bruce C. Berndt |title=The Gamma Function and the Hurwitz Zeta-Function |journal=[[The American Mathematical Monthly]] |volume=92 |number=2 |date=1985 |pages=126–130 |doi=10.2307/2322640|jstor=2322640 }}</ref><ref>{{harvnb|Whittaker|Watson|1927|p=271|loc=Section 13.21}}</ref>

==Relation to Jacobi theta function==
If <math>\vartheta (z,\tau)</math> is the Jacobi [[theta function]], then

:<math>\int_0^\infty \left[\vartheta (z,it) -1 \right] t^{s/2} \frac{dt}{t}=
\pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right)
\left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]</math>

holds for <math>\Re s > 0</math> and ''z'' complex, but not an integer. For ''z''=''n'' an integer, this simplifies to

:<math>\int_0^\infty \left[\vartheta (n,it) -1 \right] t^{s/2} \frac{dt}{t}=
2\  \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s)
=2\  \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).</math>

where ζ here is the [[Riemann zeta function]]. Note that this latter form is the [[functional equation]] for the Riemann zeta function, as originally given by Riemann. The distinction based on ''z'' being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic [[Dirac delta function|delta function]], or [[Dirac comb]] in ''z'' as <math>t\rightarrow 0</math>.

==Relation to Dirichlet ''L''-functions==
At rational arguments the Hurwitz zeta function may be expressed as a linear combination of [[Dirichlet L-function]]s and vice versa: The Hurwitz zeta function coincides with [[Riemann zeta function|Riemann's zeta function]] &zeta;(''s'') when ''a''&nbsp;=&nbsp;1, when ''a''&nbsp;=&nbsp;1/2 it is equal to (2<sup>''s''</sup>&minus;1)&zeta;(''s''),<ref name=Dav73/> and if ''a''&nbsp;=&nbsp;''n''/''k'' with ''k''&nbsp;>&nbsp;2, (''n'',''k'')&nbsp;>&nbsp;1 and 0&nbsp;<&nbsp;''n''&nbsp;<&nbsp;''k'', then<ref name=MM13>{{cite web|last=Lowry|first=David|title=Hurwitz Zeta is a sum of Dirichlet L functions, and vice-versa|url=http://mixedmath.wordpress.com/2013/02/08/hurwitz-zeta-is-a-sum-of-dirichlet-l-functions-and-vice-versa/|work=mixedmath|date=8 February 2013|access-date=8 February 2013}}</ref>

:<math>\zeta(s,n/k)=\frac{k^s}{\varphi(k)}\sum_\chi\overline{\chi}(n)L(s,\chi),</math>

the sum running over all [[Dirichlet character]]s mod ''k''. In the opposite direction we have the linear combination<ref name=Dav73/>

:<math>L(s,\chi)=\frac {1}{k^s} \sum_{n=1}^k \chi(n)\; \zeta \left(s,\frac{n}{k}\right).</math>

There is also the [[multiplication theorem]]

:<math>k^s\zeta(s)=\sum_{n=1}^k \zeta\left(s,\frac{n}{k}\right),</math>

of which a useful generalization is the ''distribution relation''<ref>{{cite book | first1=Daniel S. | last1=Kubert | author-link1=Daniel Kubert | first2=Serge | last2=Lang | author-link2=Serge Lang | title=Modular Units | series= Grundlehren der Mathematischen Wissenschaften | volume=244 | publisher=[[Springer-Verlag]] | year=1981 | isbn=0-387-90517-0 | zbl=0492.12002 | page=13 }}</ref>

:<math>\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa).</math>

(This last form is valid whenever ''q'' a natural number and 1&nbsp;&minus;&nbsp;''qa'' is not.)

==Zeros==
If ''a''=1 the Hurwitz zeta function reduces to the [[Riemann zeta function]] itself; if ''a''=1/2 it reduces to the Riemann zeta function multiplied by a simple function of the complex argument ''s'' (''vide supra''), leading in each case to the difficult study of the zeros of Riemann's zeta function. In particular, there will be no zeros with real part greater than or equal to 1. However, if 0<''a''<1 and ''a''&ne;1/2, then there are zeros of Hurwitz's zeta function in the strip 1<Re(''s'')<1+&epsilon; for any positive real number &epsilon;. This was proved by [[Harold Davenport|Davenport]] and [[Hans Heilbronn|Heilbronn]] for rational or [[Transcendental number|transcendental]] irrational ''a'',<ref>{{Citation |last1=Davenport |first1=H. |name-list-style=amp |last2=Heilbronn |first2=H. |title=On the zeros of certain Dirichlet series |journal=[[Journal of the London Mathematical Society]] |volume=11 |issue=3 |year=1936 |pages=181–185 |doi=10.1112/jlms/s1-11.3.181 | zbl=0014.21601 }}</ref> and by [[J. W. S. Cassels|Cassels]] for [[Algebraic number|algebraic]] irrational ''a''.<ref name=Dav73>Davenport (1967) p.73</ref><ref>{{Citation |last=Cassels |first=J. W. S. |title=Footnote to a note of Davenport and Heilbronn |journal=Journal of the London Mathematical Society |volume=36 |issue=1 |year=1961 |pages=177–184 |doi=10.1112/jlms/s1-36.1.177 | zbl=0097.03403 }}</ref>

==Rational values==
The Hurwitz zeta function occurs in a number of striking identities at rational values.<ref>Given by {{Citation |first1=Djurdje |last1=Cvijović |name-list-style=amp |first2=Jacek |last2=Klinowski |title=Values of the Legendre chi and Hurwitz zeta functions at rational arguments |journal=Mathematics of Computation |volume=68 |issue=228 |year=1999 |pages=1623–1630 |doi=10.1090/S0025-5718-99-01091-1|bibcode=1999MaCom..68.1623C |doi-access=free }}</ref> In particular, values in terms of the [[Euler polynomial]]s <math>E_n(x)</math>:

:<math>E_{2n-1}\left(\frac{p}{q}\right) =
(-1)^n \frac{4(2n-1)!}{(2\pi q)^{2n}}
\sum_{k=1}^q \zeta\left(2n,\frac{2k-1}{2q}\right)
\cos \frac{(2k-1)\pi p}{q}</math>

and

:<math>E_{2n}\left(\frac{p}{q}\right) =
(-1)^n \frac{4(2n)!}{(2\pi q)^{2n+1}}
\sum_{k=1}^q \zeta\left(2n+1,\frac{2k-1}{2q}\right)
\sin \frac{(2k-1)\pi p}{q}</math>

One also has

:<math>\zeta\left(s,\frac{2p-1}{2q}\right) =
2(2q)^{s-1} \sum_{k=1}^q \left[
C_s\left(\frac{k}{q}\right) \cos \left(\frac{(2p-1)\pi k}{q}\right) +
S_s\left(\frac{k}{q}\right) \sin \left(\frac{(2p-1)\pi k}{q}\right)
\right]</math>

which holds for <math>1\le p \le q</math>. Here, the <math>C_\nu(x)</math> and <math>S_\nu(x)</math> are defined by means of the [[Legendre chi function]] <math>\chi_\nu</math> as

:<math>C_\nu(x) = \operatorname{Re}\, \chi_\nu (e^{ix})</math>

and

:<math>S_\nu(x) = \operatorname{Im}\, \chi_\nu (e^{ix}).</math>

For integer values of ν, these may be expressed in terms of the Euler polynomials.  These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.

==Applications==
Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in [[number theory]], where its theory is the deepest and most developed.  However, it also occurs in the study of [[fractals]] and [[dynamical systems]]. In applied [[statistics]], it occurs in [[Zipf's law]] and the [[Zipf–Mandelbrot law]]. In [[particle physics]], it occurs in a formula by [[Julian Schwinger]],<ref>{{Citation |last=Schwinger |first=J. |title=On gauge invariance and vacuum polarization |journal=[[Physical Review]] |volume=82 |issue=5 |year=1951 |pages=664–679 |doi=10.1103/PhysRev.82.664 |bibcode=1951PhRv...82..664S}}</ref> giving an exact result for the [[pair production]] rate of a [[Paul Dirac|Dirac]] [[Dirac equation#Electromagnetic interaction|electron]] in a uniform electric field.

==Special cases and generalizations==
The Hurwitz zeta function with a positive integer ''m'' is related to the [[polygamma function]]:
:<math>\psi^{(m)}(z)= (-1)^{m+1} m! \zeta (m+1,z) \ .</math>

The [[Barnes zeta function]] generalizes the Hurwitz zeta function.

The [[Lerch transcendent]] generalizes the Hurwitz zeta:
:<math>\Phi(z, s, q) = \sum_{k=0}^\infty
\frac { z^k} {(k+q)^s}</math>
and thus
:<math>\zeta(s,a)=\Phi(1, s, a).\,</math>

[[Hypergeometric function]]

:<math>\zeta(s,a)=a^{-s}\cdot{}_{s+1}F_s(1,a_1,a_2,\ldots a_s;a_1+1,a_2+1,\ldots a_s+1;1)</math> where <math>a_1=a_2=\ldots=a_s=a\text{ and }a\notin\N\text{ and }s\in\N^+.</math>

[[Meijer G-function]]

:<math>\zeta(s,a)=G\,_{s+1,\,s+1}^{\,1,\,s+1}\left(-1 \; \left| \; \begin{matrix}0,1-a,\ldots,1-a\\0,-a,\ldots,-a\end{matrix}\right)\right.\qquad\qquad s\in\N^+.</math>

==Notes==
<references/>

==References==
*{{dlmf|id=25.11|first=T. M. |last=Apostol}}
* See chapter 12 of {{Apostol IANT}}
* Milton Abramowitz and Irene A. Stegun, ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]]'', (1964) Dover Publications, New York. {{ISBN|0-486-61272-4}}. ''(See Paragraph 6.4.10 for relationship to polygamma function.)''
* {{cite book | last=Davenport | first=Harold | author-link=Harold Davenport | title=Multiplicative number theory | publisher=Markham | series=Lectures in advanced mathematics | volume=1 | location=Chicago | year=1967 | zbl=0159.06303 }}
* {{cite journal
|first1=Jeff
|last1=Miller
|first2=Victor S.
|last2=Adamchik
|title= Derivatives of the Hurwitz Zeta Function for Rational Arguments
|journal= Journal of Computational and Applied Mathematics
|volume=100
|issue=2
|year=1998
|pages=201–206
|doi=10.1016/S0377-0427(98)00193-9
|doi-access=free
}}
* {{cite book
|title=A Course Of Modern Analysis
|title-link=A Course of Modern Analysis
|author-last1=Whittaker
|author-first1=E. T.
|author-link1=Edmund Taylor Whittaker
|author-last2=Watson
|author-first2=G. N.
|author-link2=George Neville Watson
|date=1927
|edition=4th
|publisher=[[Cambridge University Press]]
|publication-place=Cambridge, UK
}}

==External links==
* {{mathworld|urlname=HurwitzZetaFunction|title=Hurwitz Zeta Function|author=Jonathan Sondow and Eric W. Weisstein}}

[[Category:Zeta and L-functions]]