Hurwitz zeta function: Difference between revisions
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{{short description|Special function in mathematics}} |
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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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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 |
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:<math>\zeta(s,q) = \sum_{n=0}^\infty \frac{1}{(n+q)^{s}}.</math> |
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:<math>\zeta(s,a) = \sum_{n=0}^\infty \frac{1}{(n+a)^{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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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|ζ(''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> |
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[[File:Hurwitza1ov3v2.png|right|thumb|Hurwitz zeta function corresponding to {{nowrap|1=''q'' = 1/3}}. It is generated as a [[Matplotlib]] plot using a version of the [[Domain coloring]] method.<ref>http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb</ref>]] |
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[[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>]] |
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==Analytic continuation== |
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[[File:Hurwitza24ov25v2.png|right|thumb|Hurwitz zeta function corresponding to {{ |
[[File:Hurwitza24ov25v2.png|right|thumb|Hurwitz zeta function corresponding to {{math|1=''a'' = 24/25}}.]] |
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[[File:HurwitzofAz3p4j.png|right|thumb| Hurwitz zeta function as a function of {{mvar|a}} with {{math|1=''s'' = 3 + 4''i''}}.]] |
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If <math>\mathrm{Re}(s) \leq 1</math> 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]] <math>C</math> is a loop around the negative real axis. This provides an analytic continuation of <math>\zeta (s,q)</math>. |
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==Integral representation== |
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The Hurwitz zeta function can be extended by [[analytic continuation]] to a [[meromorphic function]] defined for all complex numbers <math>s</math> with <math>s \neq 1</math>. At <math>s = 1</math> it has a [[simple pole]] with [[residue (complex analysis)|residue]] <math>1</math>. The constant term is given by |
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The Hurwitz zeta function has an integral representation |
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:<math>\zeta(s,a) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}e^{-ax}}{1-e^{-x}} dx</math> |
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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 |
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:<math>\zeta(s,a)\Gamma(s) |
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= \sum_{n=0}^\infty \frac{1}{(n+a)^s} \int_0^\infty x^s e^{-x} \frac{dx}{x} |
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= \sum_{n=0}^\infty \int_0^\infty y^s e^{-(n+a)y} \frac{dy}{y}</math> |
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and then interchanging the sum and integral.<ref>{{harvnb|Apostol|1976|p=251|loc=Theorem 12.2}}</ref> |
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The integral representation above can be converted to a [[contour integral]] representation |
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:<math>\lim_{s\to 1} \left[ \zeta (s,q) - \frac{1}{s-1}\right] = |
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\ |
:<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> |
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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> |
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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> |
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where <math>\Gamma</math> is the [[gamma function]] and <math>\psi</math> is the [[digamma function]]. |
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==Hurwitz's formula== |
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==Series representation== |
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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> |
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[[File:HurwitzofAz3p4j.png|right|thumb| Hurwitz zeta function as a function of ''q'' with {{nowrap|1=''s'' = 3+4''i''}}.]] |
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:<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> |
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A convergent [[Newton 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 | jfm=56.0894.03 | url=https://eudml.org/doc/168238 }}</ref> |
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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> |
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:<math>\zeta(1-s) = \frac{2\Gamma(s)}{(2\pi)^{s}} \cos\left(\frac{\pi s}{2}\right) \zeta(s)</math> |
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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> |
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:<math>\zeta(s,q)=\frac{1}{s-1} |
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:<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> |
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\sum_{n=0}^\infty \frac{1}{n+1} |
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(for Re(''s'') < 0 and 0 < ''a'' ≤ 1). |
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\sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.</math> |
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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 |
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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>\zeta(s,a) = s \int_{-a}^\infty \frac{\lfloor x \rfloor - x + \frac{1}{2}}{(x+a)^{s+1}} dx</math> |
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(−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> |
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===Functional equation for rational ''a''=== |
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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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When ''a'' is a rational number, Hurwitz's formula leads to the following [[functional equation]]: For integers <math>1\leq m \leq n </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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Other series converging globally include these examples |
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:<math>\zeta(s,v-1)=\frac{1}{s-1}\sum_{n=0}^\infty H_{n+1}\sum_{k=0}^n (-1)^k \binom{n}{k}(k+v)^{1-s}</math> |
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:<math>\zeta(s,v)=\frac{k!}{(s-k)_k}\sum_{n=0}^\infty \frac{1}{(n+k)!}\left[{n+k \atop n}\right]\sum_{l=0}^{n+k-1}\!(-1)^l \binom{n+k-1}{l} (l+v)^{k-s},\quad k=1, 2, 3,\ldots</math> |
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:<math>\zeta(s,v)=\frac{v^{1-s}}{s-1} + \sum_{n=0}^\infty |G_{n+1}| \sum_{k=0}^{n}(-1)^k \binom{n}{k}(k+v)^{-s}</math> |
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:<math>\zeta(s,v)=\frac{(v-1)^{1-s}}{s-1} - \sum_{n=0}^\infty C_{n+1}\sum_{k=0}^{n} (-1)^k \binom{n}{k}(k+v)^{-s}</math> |
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:<math>\zeta(s,v)\big(v-\tfrac12\big)=\frac{s-2}{s-1}\zeta(s-1,v) + \sum_{n=0}^\infty (-1)^n G_{n+2}\sum_{k=0}^{n} (-1)^k \binom{n}{k} (k+v)^{-s}</math> |
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:<math>\zeta(s,v)=-\sum_{l=1}^{k-1} \frac{(k-l+1)_l}{(s-l)_l} \zeta(s-l,v) + \sum_{l=1}^{k} \frac{(k-l+1)_l}{(s-l)_l} v^{l-s} + k \sum_{n=0}^\infty (-1)^n G_{n+1}^{(k)}\sum_{k=0}^{n}(-1)^k \binom{n}{k} (k+v)^{-s}</math> |
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where {{math|''H''<sub>''n''</sub>}} are the [[Harmonic numbers]], <math>\left[{\cdot \atop \cdot}\right]</math> are the [[Stirling numbers of the first kind]], <math>(\ldots)_{\ldots}</math> is the [[Pochhammer symbol]], {{math|''G''<sub>''n''</sub>}} are the [[Gregory coefficients]], {{math|''G''{{su|b=''n''|p=(''k'')}}}} are the [[Gregory coefficients]] of higher order and {{math|''C''<sub>''n''</sub>}} are the Cauchy numbers of the second kind ({{math|''C''<sub>1</sub> {{=}} 1/2}}, {{math|''C''<sub>2</sub> {{=}} 5/12}}, {{math|''C''<sub>3</sub> {{=}} 3/8}},...), see Blagouchine's paper.<ref>{{cite journal |
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| last = Blagouchine | first = Iaroslav V. |
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| arxiv = 1606.02044 |
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| url = http://math.colgate.edu/~integers/vol18a.html |
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| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory |
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| pages = 1–45 |
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| title = Three Notes on Ser's and Hasse's Representations for the Zeta-functions |
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| volume = 18A |
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| year = 2018| bibcode = 2016arXiv160602044B}}</ref> |
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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) = |
:<math>\zeta \left(1-s,\frac{m}{n} \right) = |
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\frac{2\Gamma(s)}{ (2\pi n)^s } |
\frac{2\Gamma(s)}{ (2\pi n)^s } |
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\zeta \left( s,\frac {k}{n} \right)\right] |
\zeta \left( s,\frac {k}{n} \right)\right] |
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</math> |
</math> |
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holds for all values of ''s''. |
holds for all values of ''s''.<ref>{{harvnb|Apostol|1976|p=261|loc=Theorem 12.8}}</ref> |
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This functional equation can be written as another equivalent form: |
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<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] |
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</math>. |
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==Some finite sums== |
==Some finite sums== |
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</math> |
</math> |
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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> |
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> |
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==Series representation== |
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A convergent [[Newton series]] representation defined for (real) ''a'' > 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 | jfm=56.0894.03 |s2cid=120392534 | url=https://eudml.org/doc/168238 }}</ref> |
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:<math>\zeta(s,a)=\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} (a+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>a^{1-s}</math>; that is, |
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:<math>\Delta^n a^{1-s} = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (a+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, a) &= \frac{1}{s-1}\sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n a^{1-s}\\ |
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&= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} a^{1-s} |
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\end{align}</math> |
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==Taylor series== |
==Taylor series== |
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The derivative of the zeta in the second argument is a [[sheffer sequence|shift]]: |
The partial derivative of the zeta in the second argument is a [[sheffer sequence|shift]]: |
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:<math>\frac {\partial} {\partial |
:<math>\frac {\partial} {\partial a} \zeta (s,a) = -s\zeta(s+1,a).</math> |
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Thus, the [[Taylor series]] can be written as: |
Thus, the [[Taylor series]] can be written as: |
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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> |
:<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 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}}</ref> |
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> |
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Closely related is the '''Stark–Keiper''' formula: |
Closely related is the '''Stark–Keiper''' formula: |
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==Laurent series== |
==Laurent series== |
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The [[Laurent series]] expansion can be used to define [[Stieltjes constants]] that occur in the series |
The [[Laurent series]] expansion can be used to define generalized [[Stieltjes constants]] that occur in the series |
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:<math>\zeta(s, |
:<math>\zeta(s,a) = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(a) (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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In particular, the constant term is given by |
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==Fourier transform== |
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:<math>\lim_{s\to 1} \left[ \zeta(s,a) - \frac{1}{s-1}\right] = |
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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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\gamma_0(a)= |
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\frac{-\Gamma'(a)}{\Gamma(a)} = -\psi(a)</math> |
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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>. |
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==Discrete Fourier transform== |
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==Relation to Bernoulli polynomials== |
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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> |
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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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==Particular values== |
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In particular, the relation holds for <math>n=0</math> and one has |
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===Negative integers=== |
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:<math>\zeta(0,x)= \frac{1}{2} -x.</math> |
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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> |
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:<math>\zeta(-n,a) = -\frac{B_{n+1}(a)}{n+1}.</math> |
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For example, the <math>n=0</math> case gives<ref>{{harvnb|Apostol|1976|p=268}}</ref> |
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:<math>\zeta(0,a) = \frac{1}{2} - a.</math> |
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===''s''-derivative=== |
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The [[partial derivative]] with respect to ''s'' at ''s'' = 0 is related to the gamma function: |
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:<math>\left. \frac{\partial}{\partial s} \zeta(s,a) \right|_{s=0} = \log\Gamma(a) - \frac{1}{2} \log(2\pi)</math> |
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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> |
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==Relation to Jacobi theta function== |
==Relation to Jacobi theta function== |
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==Relation to Dirichlet ''L''-functions== |
==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 '' |
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 ''a'' = 1, when ''a'' = 1/2 it is equal to (2<sup>''s''</sup>−1)ζ(''s''),<ref name=Dav73/> and if ''a'' = ''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|date=8 February 2013|access-date=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> |
:<math>\zeta(s,n/k)=\frac{k^s}{\varphi(k)}\sum_\chi\overline{\chi}(n)L(s,\chi),</math> |
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:<math>k^s\zeta(s)=\sum_{n=1}^k \zeta\left(s,\frac{n}{k}\right),</math> |
:<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 | |
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> |
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:<math>\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa).</math> |
:<math>\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa).</math> |
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==Zeros== |
==Zeros== |
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If '' |
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''≠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 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> |
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==Rational values== |
==Rational values== |
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The Hurwitz zeta function occurs in a number of striking identities at rational values.<ref>Given by {{Citation | |
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>: |
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:<math>E_{2n-1}\left(\frac{p}{q}\right) = |
:<math>E_{2n-1}\left(\frac{p}{q}\right) = |
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The Hurwitz zeta function with a positive integer ''m'' is related to the [[polygamma function]]: |
The Hurwitz zeta function with a positive 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> |
:<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. |
The [[Barnes zeta function]] generalizes the Hurwitz zeta function. |
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\frac { z^k} {(k+q)^s}</math> |
\frac { z^k} {(k+q)^s}</math> |
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and thus |
and thus |
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:<math>\zeta |
:<math>\zeta(s,a)=\Phi(1, s, a).\,</math> |
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[[Hypergeometric function]] |
[[Hypergeometric function]] |
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*{{dlmf|id=25.11|first=T. M. |last=Apostol}} |
*{{dlmf|id=25.11|first=T. M. |last=Apostol}} |
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* See chapter 12 of {{Apostol IANT}} |
* 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 |
* 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 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 }} |
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* {{cite journal |
* {{cite journal |
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|first2=Victor S. |
|first2=Victor S. |
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|last2=Adamchik |
|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 |
|title= Derivatives of the Hurwitz Zeta Function for Rational Arguments |
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|journal= Journal of Computational and Applied Mathematics |
|journal= Journal of Computational and Applied Mathematics |
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|doi=10.1016/S0377-0427(98)00193-9 |
|doi=10.1016/S0377-0427(98)00193-9 |
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|doi-access=free |
|doi-access=free |
||
}} |
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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 |
* {{cite journal |
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|hdl=2437/90539 |
|hdl=2437/90539 |
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|hdl-access=free |
|hdl-access=free |
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}} |
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* {{cite book |
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|title=A Course Of Modern Analysis |
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|title-link=A Course of Modern Analysis |
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|author-last1=Whittaker |
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|author-first1=E. T. |
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|author-link1=Edmund Taylor Whittaker |
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|author-last2=Watson |
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|author-first2=G. N. |
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|author-link2=George Neville Watson |
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|date=1927 |
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|edition=4th |
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|publisher=[[Cambridge University Press]] |
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|publication-place=Cambridge, UK |
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}} |
}} |
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Latest revision as of 10:29, 14 August 2024
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, … by
This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s,1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882.[1]
Integral representation
[edit]The Hurwitz zeta function has an integral representation
for and (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing
and then interchanging the sum and integral.[3]
The integral representation above can be converted to a contour integral representation
where is a Hankel contour counterclockwise around the positive real axis, and the principal branch is used for the complex exponentiation . Unlike the previous integral, this integral is valid for all s, and indeed is an entire function of s.[4]
The contour integral representation provides an analytic continuation of to all . At , it has a simple pole with residue .[5]
Hurwitz's formula
[edit]The Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function:[6]
valid for Re(s) > 1 and 0 < a ≤ 1. The Riemann zeta functional equation is the special case a = 1:[7]
Hurwitz's formula can also be expressed as[8]
(for Re(s) < 0 and 0 < a ≤ 1).
Hurwitz's formula has a variety of different proofs.[9] One proof uses the contour integration representation along with the residue theorem.[6][8] A second proof uses a theta function identity, or equivalently Poisson summation.[10] These proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in Riemann's 1859 paper. Another proof of the Hurwitz formula uses Euler–Maclaurin summation to express the Hurwitz zeta function as an integral
(−1 < Re(s) < 0 and 0 < a ≤ 1) and then expanding the numerator as a Fourier series.[11]
Functional equation for rational a
[edit]When a is a rational number, Hurwitz's formula leads to the following functional equation: For integers ,
holds for all values of s.[12]
This functional equation can be written as another equivalent form:
.
Some finite sums
[edit]Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form
where m is positive integer greater than 2 and s is complex, see e.g. Appendix B in.[13]
Series representation
[edit]A convergent Newton series representation defined for (real) a > 0 and any complex s ≠ 1 was given by Helmut Hasse in 1930:[14]
This series converges uniformly on compact subsets of the s-plane to an entire function. The inner sum may be understood to be the nth forward difference of ; that is,
where Δ is the forward difference operator. Thus, one may write:
Taylor series
[edit]The partial derivative of the zeta in the second argument is a shift:
Thus, the Taylor series can be written as:
Alternatively,
with .[15]
Closely related is the Stark–Keiper formula:
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
[edit]The Laurent series expansion can be used to define generalized Stieltjes constants that occur in the series
In particular, the constant term is given by
where is the gamma function and is the digamma function. As a special case, .
Discrete Fourier transform
[edit]The discrete Fourier transform of the Hurwitz zeta function with respect to the order s is the Legendre chi function.[16]
Particular values
[edit]Negative integers
[edit]The values of ζ(s, a) at s = 0, −1, −2, ... are related to the Bernoulli polynomials:[17]
For example, the case gives[18]
s-derivative
[edit]The partial derivative with respect to s at s = 0 is related to the gamma function:
In particular, The formula is due to Lerch.[19][20]
Relation to Jacobi theta function
[edit]If is the Jacobi theta function, then
holds for and z complex, but not an integer. For z=n an integer, this simplifies to
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 delta function, or Dirac comb in z as .
Relation to Dirichlet L-functions
[edit]At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's zeta function ζ(s) when a = 1, when a = 1/2 it is equal to (2s−1)ζ(s),[21] and if a = n/k with k > 2, (n,k) > 1 and 0 < n < k, then[22]
the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination[21]
There is also the multiplication theorem
of which a useful generalization is the distribution relation[23]
(This last form is valid whenever q a natural number and 1 − qa is not.)
Zeros
[edit]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≠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 Davenport and Heilbronn for rational or transcendental irrational a,[24] and by Cassels for algebraic irrational a.[21][25]
Rational values
[edit]The Hurwitz zeta function occurs in a number of striking identities at rational values.[26] In particular, values in terms of the Euler polynomials :
and
One also has
which holds for . Here, the and are defined by means of the Legendre chi function as
and
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
[edit]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,[27] giving an exact result for the pair production rate of a Dirac electron in a uniform electric field.
Special cases and generalizations
[edit]The Hurwitz zeta function with a positive integer m is related to the polygamma function:
The Barnes zeta function generalizes the Hurwitz zeta function.
The Lerch transcendent generalizes the Hurwitz zeta:
and thus
- where
Notes
[edit]- ^ Hurwitz, Adolf (1882). "Einige Eigenschaften der Dirichlet'schen Functionen , die bei der Bestimmung der Classenanzahlen binärer quadratischer Formen auftreten". Zeitschrift für Mathematik und Physik (in German). 27: 86–101.
- ^ "Jupyter Notebook Viewer".
- ^ Apostol 1976, p. 251, Theorem 12.2
- ^ Whittaker & Watson 1927, p. 266, Section 13.13
- ^ Apostol 1976, p. 255, Theorem 12.4
- ^ a b Apostol 1976, p. 257, Theorem 12.6
- ^ Apostol 1976, p. 259, Theorem 12.7
- ^ a b Whittaker & Watson 1927, pp. 268–269, Section 13.15
- ^ See the references in Section 4 of: Kanemitsu, S.; Tanigawa, Y.; Tsukada, H.; Yoshimoto, M. (2007). "Contributions to the theory of the Hurwitz zeta-function". Hardy-Ramanujan Journal. 30: 31–55. doi:10.46298/hrj.2007.159. Zbl 1157.11036.
- ^ Fine, N. J. (June 1951). "Note on the Hurwitz Zeta-Function". Proceedings of the American Mathematical Society. 2 (3): 361–364. doi:10.2307/2031757. JSTOR 2031757. Zbl 0043.07802.
- ^ Berndt, Bruce C. (Winter 1972). "On the Hurwitz zeta-function". Rocky Mountain Journal of Mathematics. 2 (1): 151–158. doi:10.1216/RMJ-1972-2-1-151. Zbl 0229.10023.
- ^ Apostol 1976, p. 261, Theorem 12.8
- ^ Blagouchine, I.V. (2014). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148. Elsevier: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009.
- ^ Hasse, Helmut (1930), "Ein Summierungsverfahren für die Riemannsche ζ-Reihe", Mathematische Zeitschrift, 32 (1): 458–464, doi:10.1007/BF01194645, JFM 56.0894.03, S2CID 120392534
- ^ Vepstas, Linas (2007). "An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions". Numerical Algorithms. 47 (3): 211–252. arXiv:math/0702243. Bibcode:2008NuAlg..47..211V. doi:10.1007/s11075-007-9153-8. S2CID 15131811.
- ^ Jacek Klinowski, Djurdje Cvijović (1999). "Values of the Legendre chi and Hurwitz zeta functions at rational arguments". Mathematics of Computation. 68 (228): 1623–1631. Bibcode:1999MaCom..68.1623C. doi:10.1090/S0025-5718-99-01091-1.
- ^ Apostol 1976, p. 264, Theorem 12.13
- ^ Apostol 1976, p. 268
- ^ Berndt, Bruce C. (1985). "The Gamma Function and the Hurwitz Zeta-Function". The American Mathematical Monthly. 92 (2): 126–130. doi:10.2307/2322640. JSTOR 2322640.
- ^ Whittaker & Watson 1927, p. 271, Section 13.21
- ^ a b c Davenport (1967) p.73
- ^ Lowry, David (8 February 2013). "Hurwitz Zeta is a sum of Dirichlet L functions, and vice-versa". mixedmath. Retrieved 8 February 2013.
- ^ Kubert, Daniel S.; Lang, Serge (1981). Modular Units. Grundlehren der Mathematischen Wissenschaften. Vol. 244. Springer-Verlag. p. 13. ISBN 0-387-90517-0. Zbl 0492.12002.
- ^ Davenport, H. & Heilbronn, H. (1936), "On the zeros of certain Dirichlet series", Journal of the London Mathematical Society, 11 (3): 181–185, doi:10.1112/jlms/s1-11.3.181, Zbl 0014.21601
- ^ Cassels, J. W. S. (1961), "Footnote to a note of Davenport and Heilbronn", Journal of the London Mathematical Society, 36 (1): 177–184, doi:10.1112/jlms/s1-36.1.177, Zbl 0097.03403
- ^ Given by Cvijović, Djurdje & Klinowski, Jacek (1999), "Values of the Legendre chi and Hurwitz zeta functions at rational arguments", Mathematics of Computation, 68 (228): 1623–1630, Bibcode:1999MaCom..68.1623C, doi:10.1090/S0025-5718-99-01091-1
- ^ Schwinger, J. (1951), "On gauge invariance and vacuum polarization", Physical Review, 82 (5): 664–679, Bibcode:1951PhRv...82..664S, doi:10.1103/PhysRev.82.664
References
[edit]- Apostol, T. M. (2010), "Hurwitz zeta function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
- See chapter 12 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
- Milton Abramowitz and Irene A. 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.)
- Davenport, Harold (1967). Multiplicative number theory. Lectures in advanced mathematics. Vol. 1. Chicago: Markham. Zbl 0159.06303.
- Miller, Jeff; Adamchik, Victor S. (1998). "Derivatives of the Hurwitz Zeta Function for Rational Arguments". Journal of Computational and Applied Mathematics. 100 (2): 201–206. doi:10.1016/S0377-0427(98)00193-9.
- Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory. 130 (2): 360–369. doi:10.1016/j.jnt.2009.08.005. hdl:2437/90539.
- Whittaker, E. T.; Watson, G. N. (1927). A Course Of Modern Analysis (4th ed.). Cambridge, UK: Cambridge University Press.
External links
[edit]- Jonathan Sondow and Eric W. Weisstein. "Hurwitz Zeta Function". MathWorld.