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| In [[mathematics]], the '''Hurwitz zeta function''', named after [[Adolf Hurwitz]], is one of the many [[zeta function]]s. It is formally defined for [[complex number|complex]] arguments ''s'' with Re(''s'') > 1 and ''q'' with Re(''q'') > 0 by
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| :<math>\zeta(s,q) = \sum_{n=0}^\infty \frac{1}{(q+n)^{s}}.</math>
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| This series is [[absolutely convergent]] for the given values of ''s'' and ''q'' and can be extended to a [[meromorphic function]] defined for all ''s''≠1. The [[Riemann zeta function]] is ζ(''s'',1).
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| ==Analytic continuation==
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| If Re(s) ≤ 1 the Hurwitz zeta function can be defined by the equation
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| :<math>\zeta (s,q)=\Gamma(1-s)\frac{1}{2 \pi i} \int_C \frac{z^{s-1}e^{qz}}{1-e^z}dz</math>
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| where the [[Contour integration|contour]] C is a loop around the negative real axis. This provides an analytic continuation of <math>\zeta (s,q)</math>.
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| The Hurwitz zeta function can be extended by [[analytic continuation]] to a [[meromorphic function]] defined for all complex numbers ''s'' with ''s'' ≠ 1. At ''s'' = 1 it has a [[simple pole]] with [[residue (complex analysis)|residue]] 1. The constant term is given by
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| :<math>\lim_{s\to 1} \left[ \zeta (s,q) - \frac{1}{s-1}\right] =
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| \frac{-\Gamma'(q)}{\Gamma(q)} = -\psi(q)</math>
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| where Γ is the [[Gamma function]] and ψ is the [[digamma function]].
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| ==Series representation==
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| A convergent series representation defined for (real) ''q'' > 0 and any complex ''s'' ≠ 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 }}</ref>
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| :<math>\zeta(s,q)=\frac{1}{s-1}
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| \sum_{n=0}^\infty \frac{1}{n+1}
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| \sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.</math>
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| 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>q^{1-s}</math>; that is,
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| :<math>\Delta^n q^{1-s} = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (q+k)^{1-s}</math>
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| where Δ is the [[forward difference operator]]. Thus, one may write
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| :<math>\begin{align}
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| \zeta(s, q) &= \frac{1}{s-1}\sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n q^{1-s}\\
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| &= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} q^{1-s}
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| \end{align}</math>
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| ==Integral representation==
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| The function has an integral representation in terms of the [[Mellin transform]] as
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| :<math>\zeta(s,q)=\frac{1}{\Gamma(s)} \int_0^\infty
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| \frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt</math>
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| for <math>\Re s>1</math> and <math>\Re q >0.</math>
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| ==Hurwitz's formula==
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| Hurwitz's formula is the theorem that
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| :<math>\zeta(1-s,x)=\frac{1}{2s}\left[e^{-i\pi s/2}\beta(x;s) + e^{i\pi s/2} \beta(1-x;s) \right]</math>
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| where
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| :<math>\beta(x;s)=
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| 2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}=
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| \frac{2\Gamma(s+1)}{(2\pi)^s} \mbox{Li}_s (e^{2\pi ix})
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| </math>
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| is a representation of the zeta that is valid for <math>0\le x\le 1</math> and s > 1. Here, <math>\text{Li}_s (z)</math> is the [[polylogarithm]].
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| ==Functional equation==
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| The [[functional equation]] relates values of the zeta on the left- and right-hand sides of the complex plane. For integers <math>1\leq m \leq n </math>,
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| :<math>\zeta \left(1-s,\frac{m}{n} \right) =
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| \frac{2\Gamma(s)}{ (2\pi n)^s }
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| \sum_{k=1}^n \left[\cos
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| \left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\;
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| \zeta \left( s,\frac {k}{n} \right)\right]
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| </math>
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| holds for all values of ''s''.
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| ==Taylor series==
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| The derivative of the zeta in the second argument is a [[sheffer sequence|shift]]:
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| :<math>\frac {\partial} {\partial q} \zeta (s,q) = -s\zeta(s+1,q).</math>
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| Thus, the [[Taylor series]] has the distinctly [[umbral calculus|umbral]] form:
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| :<math>\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!}
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| \frac {\partial^k} {\partial x^k} \zeta (s,x) =
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| \sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).</math>
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| Alternatively,
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| :<math>\zeta(s, q) = \frac{1}{q^s} + \sum_{n=0}^{\infty} (-q)^n {s + n - 1 \choose n} \zeta(s + n),</math>
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| with <math>|q| < 1</math>.<ref>{{cite arXiv |last=Vepstas |first=Linas |authorlink= |eprint=math/0702243 |title=An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions |class= |year=2007 |accessdate=29 August 2013 }}</ref>
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| Closely related is the '''Stark–Keiper''' formula:
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| :<math>\zeta(s,N) =
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| \sum_{k=0}^\infty \left[ N+\frac {s-1}{k+1}\right]
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| {s+k-1 \choose s-1} (-1)^k \zeta (s+k,N) </math>
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| which holds for integer ''N'' and arbitrary ''s''. See also [[Faulhaber's formula]] for a similar relation on finite sums of powers of integers.
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| ==Laurent series==
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| The [[Laurent series]] expansion can be used to define [[Stieltjes constants]] that occur in the series
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| :<math>\zeta(s,q)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(q) \; (s-1)^n.</math>
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| Specifically <math>\gamma_0(q) = -\psi(q)</math> and <math>\gamma_0(1) = -\psi(1) = \gamma_0 = \gamma</math>.
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| ==Fourier transform==
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| The [[discrete Fourier transform]] of the Hurwitz zeta function with respect to the order ''s'' is the [[Legendre chi function]].
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| ==Relation to Bernoulli polynomials==
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| The function <math>\beta</math> defined above generalizes the [[Bernoulli polynomials]]:
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| :<math>B_n(x) = -\Re \left[ (-i)^n \beta(x;n) \right] </math>
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| where <math>\Re z</math> denotes the real part of ''z''. Alternately,
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| :<math>\zeta(-n,x)=-{B_{n+1}(x) \over n+1}.</math>
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| In particular, the relation holds for <math>n=0</math> and one has
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| :<math>\zeta(0,x)= \frac{1}{2} -x.</math>
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| ==Relation to Jacobi theta function==
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| If <math>\vartheta (z,\tau)</math> is the Jacobi [[theta function]], then
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| :<math>\int_0^\infty \left[\vartheta (z,it) -1 \right] t^{s/2} \frac{dt}{t}=
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| \pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right)
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| \left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]</math>
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| holds for <math>\Re s > 0</math> and ''z'' complex, but not an integer. For ''z''=''n'' an integer, this simplifies to
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| :<math>\int_0^\infty \left[\vartheta (n,it) -1 \right] t^{s/2} \frac{dt}{t}=
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| 2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s)
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| =2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).</math>
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| 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 [[Dirac delta function]] in ''z'' as <math>t\rightarrow 0</math>.
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| ==Relation to Dirichlet ''L''-functions==
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| 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]] ζ(''s'') when ''q'' = 1, when ''q'' = 1/2 it is equal to (2<sup>''s''</sup>−1)ζ(''s''),<ref name=Dav73/> and if ''q'' = ''n''/''k'' with ''k'' > 2, (''n'',''k'') > 1 and 0 < ''n'' < ''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|accessdate=8 February 2013}}</ref>
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| :<math>\zeta(s,n/k)=\frac{k^s}{\varphi(k)}\sum_\chi\overline{\chi}(n)L(s,\chi),</math>
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| the sum running over all [[Dirichlet character]]s mod ''k''. In the opposite direction we have the linear combination<ref name=Dav73/>
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| :<math>L(s,\chi)=\frac {1}{k^s} \sum_{n=1}^k \chi(n)\; \zeta \left(s,\frac{n}{k}\right).</math>
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| There is also the [[multiplication theorem]]
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| :<math>k^s\zeta(s)=\sum_{n=1}^k \zeta\left(s,\frac{n}{k}\right),</math>
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| of which a useful generalization is the ''distribution relation''<ref>{{cite book | first1=Daniel S. | last1=Kubert | authorlink1=Daniel Kubert | first2=Serge | last2=Lang | authorlink2=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>
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| :<math>\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa).</math>
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| (This last form is valid whenever ''q'' a natural number and 1 − ''qa'' is not.)
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| ==Zeros==
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| If ''q''=1 the Hurwitz zeta function reduces to the [[Riemann zeta function]] itself; if ''q''=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<''q''<1 and ''q''≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1<Re(''s'')<1+ε for any positive real number ε. This was proved by [[Harold Davenport|Davenport]] and [[Hans Heilbronn|Heilbronn]] for rational and non-algebraic irrational ''q'',<ref>{{Citation |last=Davenport |first=H. |lastauthoramp=yes |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 }}</ref> and by [[J. W. S. Cassels|Cassels]] for algebraic irrational ''q''.<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 }}</ref><ref name=Dav73>Davenport (1967) p.73</ref>
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| ==Rational values==
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| The Hurwitz zeta function occurs in a number of striking identities at rational values.<ref>Given by {{Citation |first=Djurdje |last=Cvijović |lastauthoramp=yes |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 }}</ref> In particular, values in terms of the [[Euler polynomial]]s <math>E_n(x)</math>:
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| :<math>E_{2n-1}\left(\frac{p}{q}\right) =
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| (-1)^n \frac{4(2n-1)!}{(2\pi q)^{2n}}
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| \sum_{k=1}^q \zeta\left(2n,\frac{2k-1}{2q}\right)
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| \cos \frac{(2k-1)\pi p}{q}</math>
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| and
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| :<math>E_{2n}\left(\frac{p}{q}\right) =
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| (-1)^n \frac{4(2n)!}{(2\pi q)^{2n+1}}
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| \sum_{k=1}^q \zeta\left(2n+1,\frac{2k-1}{2q}\right)
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| \sin \frac{(2k-1)\pi p}{q}</math>
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| One also has
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| :<math>\zeta\left(s,\frac{2p-1}{2q}\right) =
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| 2(2q)^{s-1} \sum_{k=1}^q \left[
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| C_s\left(\frac{k}{q}\right) \cos \left(\frac{(2p-1)\pi k}{q}\right) +
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| S_s\left(\frac{k}{q}\right) \sin \left(\frac{(2p-1)\pi k}{q}\right)
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| \right]</math>
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| 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
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| :<math>C_\nu(x) = \operatorname{Re}\, \chi_\nu (e^{ix})</math>
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| and
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| :<math>S_\nu(x) = \operatorname{Im}\, \chi_\nu (e^{ix}).</math>
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| 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.
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| ==Applications==
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| 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.
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| ==Special cases and generalizations==
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| The Hurwitz zeta function with non-negative integer ''m'' is related to the [[polygamma function]]:
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| :<math>\psi^{(m)}(z)= (-1)^{m+1} m! \zeta (m+1,z) \ .</math>
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| For negative integer −''n'' the values are related to the [[Bernoulli polynomials]]:<ref name=Ap264>Apostol (1976) p.264</ref>
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| :<math>\zeta(-n,x) = - \frac{B_{n+1}(x)}{n+1} \ . </math>
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| The [[Barnes zeta function]] generalizes the Hurwitz zeta function.
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| The [[Lerch transcendent]] generalizes the Hurwitz zeta:
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| :<math>\Phi(z, s, q) = \sum_{k=0}^\infty
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| \frac { z^k} {(k+q)^s}</math>
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| and thus
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| :<math>\zeta (s,q)=\Phi(1, s, q).\,</math>
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| [[Hypergeometric function]]
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| :<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> | |
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| [[Meijer G-function]]
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| :<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>
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| ==Notes==
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| <references/> | |
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| ==References==
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| *{{dlmf|id=25.11|first=T. M. |last=Apostol}}
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| * See chapter 12 of {{Apostol IANT}}
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| * Milton Abramowitz and Irene A. Stegun, ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]]'', (1964) Dover Publications, New York. ISBN 0-486-61272-4. ''(See [http://www.math.sfu.ca/~cbm/aands/page_260.htm Paragraph 6.4.10] for relationship to polygamma function.)''
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| * {{cite book | last=Davenport | first=Harold | authorlink=Harold Davenport | title=Multiplicative number theory | publisher=Markham | series=Lectures in advanced mathematics | volume=1 | location=Chicago | year=1967 | zbl=0159.06303 }}
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| * {{cite journal
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| |first1=Jeff
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| |last1=Miller
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| |first2=Victor S.
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| |last2=Adamchik
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| |url=http://www-2.cs.cmu.edu/~adamchik/articles/hurwitz.htm
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| |title= Derivatives of the Hurwitz Zeta Function for Rational Arguments
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| |journal= Journal of Computational and Applied Mathematics
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| |volume=100
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| |year=1998
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| |pages=201–206
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| |doi=10.1016/S0377-0427(98)00193-9
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| }}
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| * {{cite web
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| |first1=Linas
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| |last1=Vepstas
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| |url=http://www.linas.org/math/gkw.pdf
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| |title= The Bernoulli Operator, the Gauss–Kuzmin–Wirsing Operator, and the Riemann Zeta
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| }}
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| * {{cite journal
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| |first1=István
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| |last1=Mező
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| |first2=Ayhan
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| |last2=Dil
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| |doi=10.1016/j.jnt.2009.08.005
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| |title=Hyperharmonic series involving Hurwitz zeta function
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| |journal=Journal of Number Theory
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| |year=2010
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| |volume=130
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| |number=2
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| |pages=360–369
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| }}
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| ==External links==
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| * {{mathworld|urlname=HurwitzZetaFunction|title=Hurwitz Zeta Function|author=Jonathan Sondow and Eric W. Weisstein}}
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| [[Category:Zeta and L-functions]]
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