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''Epstein <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e1201302.png" />-function''
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''Epstein $\zeta$-function''
  
A function belonging to a class of [[Dirichlet series|Dirichlet series]] generalizing the [[Riemann zeta-function|Riemann zeta-function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e1201303.png" /> (cf. also [[Zeta-function|Zeta-function]]). It was introduced by P. Epstein [[#References|[a4]]] in 1903 after special cases had been dealt with by L. Kronecker [[#References|[a6]]], IV, 495. Given a real positive-definite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e1201304.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e1201305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e1201306.png" />, the Epstein zeta-function is defined by
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A function belonging to a class of [[Dirichlet series]] generalizing the [[Riemann zeta-function]] $\zeta(s)$ (cf. also [[Zeta-function]]). It was introduced by P. Epstein [[#References|[a4]]] in 1903 after special cases had been dealt with by L. Kronecker [[#References|[a6]]], IV, 495. Given a real positive-definite $n\times n)$-matrix $T$ and $s \in \mathbf{C}$, the Epstein zeta-function is defined by
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e1201307.png" /></td> </tr></table>
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\zeta(T;s) = \sum_{\mathbf{0} \ne g \in \mathbf{Z}^n} (g^\top T g)^{-s}
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e1201308.png" /> stands for the transpose of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e1201309.png" />. The series converges absolutely for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013012.png" />, it equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013013.png" />.
+
where $g^\top$ stands for the transpose of $g$. The series converges absolutely for $\mathrm{re} s > n/2$. If $n=1$ and $T=(1)$, it equals $2\zeta(2s)$.
  
 
The Epstein zeta-function shares many properties with the Riemann zeta-function (cf. [[#References|[a5]]], V.Sect. 5, [[#References|[a8]]], 1.4, [[#References|[a9]]]):
 
The Epstein zeta-function shares many properties with the Riemann zeta-function (cf. [[#References|[a5]]], V.Sect. 5, [[#References|[a8]]], 1.4, [[#References|[a9]]]):
 +
$$
 +
\xi(T;s) = \pi^{-s} \Gamma(s) \zeta(T;s)
 +
$$
 +
possesses a meromorphic continuation to the whole $s$-plane (cf. also [[Analytic continuation]]) with two simple poles, at $s = n/2$ and $s=0$, and satisfies the functional equation
 +
$$
 +
\xi(T;s) = (\det T)^{-1/2} \xi\left({ T^{-1};\frac{n}{2}-s }\right) \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013014.png" /></td> </tr></table>
+
Thus, $\zeta(T;s)$ is holomorphic in $s \in \mathbf{C}$ except for a simple pole at $s=n/2$ with residue
 
+
$$
possesses a meromorphic continuation to the whole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013015.png" />-plane (cf. also [[Analytic continuation|Analytic continuation]]) with two simple poles, at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013017.png" />, and satisfies the functional equation
+
\frac{\pi^{n/2}}{ \Gamma(n/2)\sqrt{\det T} } \ .
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013018.png" /></td> </tr></table>
 
 
 
Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013019.png" /> is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013020.png" /> except for a simple pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013021.png" /> with residue
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013022.png" /></td> </tr></table>
 
  
 
Moreover, one has
 
Moreover, one has
 +
$$
 +
\zeta(T;0) = -1
 +
$$
 +
$$
 +
\zeta(T;-m) = 0\ \ \text{for}\ \ m=1,2,\ldots \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013023.png" /></td> </tr></table>
+
It should be noted that the behaviour may be totally different from the Riemann zeta-function. For instance, for $n>1$ there exist matrices $T$ such that $\zeta(T;s)$ has infinitely many zeros in the half-plane of absolute convergence (cf. [[#References|[a1]]]), respectively a zero in any point of the real interval $(0,n/2)$ (cf. [[#References|[a8]]], 4.4).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013024.png" /></td> </tr></table>
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The Epstein zeta-function is an [[automorphic form]] for the [[unimodular group]] $\mathrm{GL}_n(\mathbf{Z})$ (cf. [[#References|[a8]]], 4.5), i.e.
 +
$$
 +
\zeta(U^\top T u;s) = \zeta(T;s) \ \ text{for}\ \ U \in \mathrm{GL}_n(\mathbf{Z}) \ .
 +
$$
  
It should be noted that the behaviour may be totally different from the Riemann zeta-function. For instance, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013025.png" /> there exist matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013027.png" /> has infinitely many zeros in the half-plane of absolute convergence (cf. [[#References|[a1]]]), respectively a zero in any point of the real interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013028.png" /> (cf. [[#References|[a8]]], 4.4).
+
It has a Fourier expansion in the partial Iwasawa coordinates of $T$ involving [[Bessel functions]] (cf. [[#References|[a8]]], 4.5). For $n=2$ it coincides with the real-analytic [[Eisenstein series]] on the upper half-plane (cf. [[Modular form]]; [[#References|[a5]]], V.Sect. 5, [[#References|[a8]]], 3.5).
  
The Epstein zeta-function is an [[Automorphic form|automorphic form]] for the [[Unimodular group|unimodular group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013029.png" /> (cf. [[#References|[a8]]], 4.5), i.e.
+
The Epstein zeta-function can also be described in terms of a lattice $\Lambda = \mathbf{Z}\lambda_1 + \cdots + \mathbf{Z}\lambda_n$ in an $n$-dimensional Euclidean vector space $(V,\sigma)$. One has
 +
$$
 +
\zeta(T;s) = \sum_{0 /ne \lambda \in \Lambda} \sigma(\lambda,\lambda)^{-s} \ ,
 +
$$
 +
where $T = (\sigma(\lambda_i,\lambda_j))$ is the [[Gram matrix]] of the basis $\lambda_1,\ldots,\lambda_n$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013030.png" /></td> </tr></table>
+
Moreover, the Epstein zeta-function is related with number-theoretical problems. It is involved in the investigation of the  "class number one problem" for imaginary quadratic number fields (cf. [[#References|[a7]]]). In the case of an arbitrary algebraic [[number field]] it gives an integral representation of the associated [[Dedekind zeta-function]] (cf. [[#References|[a8]]], 1.4).
  
It has a Fourier expansion in the partial Iwasawa coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013031.png" /> involving [[Bessel functions|Bessel functions]] (cf. [[#References|[a8]]], 4.5). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013032.png" /> it coincides with the real-analytic Eisenstein series on the upper half-plane (cf. [[Modular form|Modular form]]; [[#References|[a5]]], V.Sect. 5, [[#References|[a8]]], 3.5).
+
The Epstein zeta-function plays an important role in crystallography, e.g. in the determination of the Madelung constant (cf. [[#References|[a8]]], 1.4). Moreover, there are several applications in mathematical physics, e.g. [[quantum field theory]] and the Wheeler–DeWitt equation (cf. [[#References|[a2]]], [[#References|[a3]]]).
  
The Epstein zeta-function can also be described in terms of a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013033.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013034.png" />-dimensional Euclidean vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013035.png" />. One has
+
====References====
 
+
<table>
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013036.png" /></td> </tr></table>
+
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Davenport,  H. Heilbronn,  "On the zeros of certain Dirichlet series I, II" ''J. London Math. Soc.'' , '''11'''  (1936)  pp. 181–185; 307–312</TD></TR>
 
+
<TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Elizalde,  "Ten physical applications of spectral zeta functions" , ''Lecture Notes Physics'' , Springer  (1995)</TD></TR>
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013037.png" /> is the [[Gram matrix|Gram matrix]] of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e120/e120130/e12013038.png" />.
+
<TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Elizalde,  "Multidimensional extension of the generalized Chowla–Selberg formula" ''Comm. Math. Phys.'' , '''198'''  (1998)  pp. 83–95</TD></TR>
 
+
<TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Epstein,  "Zur Theorie allgemeiner Zetafunktionen I, II" ''Math. Ann.'' , '''56/63'''  (1903/7)  pp. 615–644; 205–216</TD></TR>
Moreover, the Epstein zeta-function is related with number-theoretical problems. It is involved in the investigation of the  "class number one problemfor imaginary quadratic number fields (cf. [[#References|[a7]]]). In the case of an arbitrary algebraic [[Number field|number field]] it gives an integral representation of the attached [[Dedekind zeta-function|Dedekind zeta-function]] (cf. [[#References|[a8]]], 1.4).
+
<TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Koecher,  A. Krieg,  "Elliptische Funktionen und Modulformen" , Springer  (1998)</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Kronecker,  "Werke I—V" , Chelsea  (1968)</TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Selberg,  Chowla, S.,  "On Epstein's Zeta-function" ''J. Reine Angew. Math.'' , '''227'''  (1967)  pp. 86–110</TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Terras,  "Harmonic analysis on symmetric spaces and applications" , '''I, II''' , Springer  (1985/8)</TD></TR>
 +
<TR><TD valign="top">[a9]</TD> <TD valign="top"> E.C. Titchmarsh,  D.R. Heath–Brown,  "The theory of the Riemann zeta-function" , Clarendon Press  (1986)</TD></TR>
 +
</table>
  
The Epstein zeta-function plays an important role in crystallography, e.g. in the determination of the Madelung constant (cf. [[#References|[a8]]], 1.4). Moreover, there are several applications in mathematical physics, e.g. [[Quantum field theory|quantum field theory]] and the Wheeler–DeWitt equation (cf. [[#References|[a2]]], [[#References|[a3]]]).
+
{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Davenport,  H. Heilbronn,  "On the zeros of certain Dirichlet series I, II"  ''J. London Math. Soc.'' , '''11'''  (1936)  pp. 181–185; 307–312</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Elizalde,  "Ten physical applications of spectral zeta functions" , ''Lecture Notes Physics'' , Springer  (1995)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Elizalde,  "Multidimensional extension of the generalized Chowla–Selberg formula"  ''Comm. Math. Phys.'' , '''198'''  (1998)  pp. 83–95</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Epstein,  "Zur Theorie allgemeiner Zetafunktionen I, II"  ''Math. Ann.'' , '''56/63'''  (1903/7)  pp. 615–644; 205–216</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Koecher,  A. Krieg,  "Elliptische Funktionen und Modulformen" , Springer  (1998)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Kronecker,  "Werke I—V" , Chelsea  (1968)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Selberg,  Chowla, S.,  "On Epstein's Zeta-function"  ''J. Reine Angew. Math.'' , '''227'''  (1967)  pp. 86–110</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A. Terras,  "Harmonic analysis on symmetric spaces and applications" , '''I, II''' , Springer  (1985/8)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.C. Titchmarsh,  D.R. Heath–Brown,  "The theory of the Riemann zeta-function" , Clarendon Press  (1986)</TD></TR></table>
 

Revision as of 18:33, 5 October 2017

Epstein $\zeta$-function

A function belonging to a class of Dirichlet series generalizing the Riemann zeta-function $\zeta(s)$ (cf. also Zeta-function). It was introduced by P. Epstein [a4] in 1903 after special cases had been dealt with by L. Kronecker [a6], IV, 495. Given a real positive-definite $n\times n)$-matrix $T$ and $s \in \mathbf{C}$, the Epstein zeta-function is defined by $$ \zeta(T;s) = \sum_{\mathbf{0} \ne g \in \mathbf{Z}^n} (g^\top T g)^{-s} $$ where $g^\top$ stands for the transpose of $g$. The series converges absolutely for $\mathrm{re} s > n/2$. If $n=1$ and $T=(1)$, it equals $2\zeta(2s)$.

The Epstein zeta-function shares many properties with the Riemann zeta-function (cf. [a5], V.Sect. 5, [a8], 1.4, [a9]): $$ \xi(T;s) = \pi^{-s} \Gamma(s) \zeta(T;s) $$ possesses a meromorphic continuation to the whole $s$-plane (cf. also Analytic continuation) with two simple poles, at $s = n/2$ and $s=0$, and satisfies the functional equation $$ \xi(T;s) = (\det T)^{-1/2} \xi\left({ T^{-1};\frac{n}{2}-s }\right) \ . $$

Thus, $\zeta(T;s)$ is holomorphic in $s \in \mathbf{C}$ except for a simple pole at $s=n/2$ with residue $$ \frac{\pi^{n/2}}{ \Gamma(n/2)\sqrt{\det T} } \ . $$

Moreover, one has $$ \zeta(T;0) = -1 $$ $$ \zeta(T;-m) = 0\ \ \text{for}\ \ m=1,2,\ldots \ . $$

It should be noted that the behaviour may be totally different from the Riemann zeta-function. For instance, for $n>1$ there exist matrices $T$ such that $\zeta(T;s)$ has infinitely many zeros in the half-plane of absolute convergence (cf. [a1]), respectively a zero in any point of the real interval $(0,n/2)$ (cf. [a8], 4.4).

The Epstein zeta-function is an automorphic form for the unimodular group $\mathrm{GL}_n(\mathbf{Z})$ (cf. [a8], 4.5), i.e. $$ \zeta(U^\top T u;s) = \zeta(T;s) \ \ text{for}\ \ U \in \mathrm{GL}_n(\mathbf{Z}) \ . $$

It has a Fourier expansion in the partial Iwasawa coordinates of $T$ involving Bessel functions (cf. [a8], 4.5). For $n=2$ it coincides with the real-analytic Eisenstein series on the upper half-plane (cf. Modular form; [a5], V.Sect. 5, [a8], 3.5).

The Epstein zeta-function can also be described in terms of a lattice $\Lambda = \mathbf{Z}\lambda_1 + \cdots + \mathbf{Z}\lambda_n$ in an $n$-dimensional Euclidean vector space $(V,\sigma)$. One has $$ \zeta(T;s) = \sum_{0 /ne \lambda \in \Lambda} \sigma(\lambda,\lambda)^{-s} \ , $$ where $T = (\sigma(\lambda_i,\lambda_j))$ is the Gram matrix of the basis $\lambda_1,\ldots,\lambda_n$.

Moreover, the Epstein zeta-function is related with number-theoretical problems. It is involved in the investigation of the "class number one problem" for imaginary quadratic number fields (cf. [a7]). In the case of an arbitrary algebraic number field it gives an integral representation of the associated Dedekind zeta-function (cf. [a8], 1.4).

The Epstein zeta-function plays an important role in crystallography, e.g. in the determination of the Madelung constant (cf. [a8], 1.4). Moreover, there are several applications in mathematical physics, e.g. quantum field theory and the Wheeler–DeWitt equation (cf. [a2], [a3]).

References

[a1] H. Davenport, H. Heilbronn, "On the zeros of certain Dirichlet series I, II" J. London Math. Soc. , 11 (1936) pp. 181–185; 307–312
[a2] E. Elizalde, "Ten physical applications of spectral zeta functions" , Lecture Notes Physics , Springer (1995)
[a3] E. Elizalde, "Multidimensional extension of the generalized Chowla–Selberg formula" Comm. Math. Phys. , 198 (1998) pp. 83–95
[a4] P. Epstein, "Zur Theorie allgemeiner Zetafunktionen I, II" Math. Ann. , 56/63 (1903/7) pp. 615–644; 205–216
[a5] M. Koecher, A. Krieg, "Elliptische Funktionen und Modulformen" , Springer (1998)
[a6] L. Kronecker, "Werke I—V" , Chelsea (1968)
[a7] A. Selberg, Chowla, S., "On Epstein's Zeta-function" J. Reine Angew. Math. , 227 (1967) pp. 86–110
[a8] A. Terras, "Harmonic analysis on symmetric spaces and applications" , I, II , Springer (1985/8)
[a9] E.C. Titchmarsh, D.R. Heath–Brown, "The theory of the Riemann zeta-function" , Clarendon Press (1986)
How to Cite This Entry:
Epstein zeta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Epstein_zeta-function&oldid=16021
This article was adapted from an original article by A. Krieg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article