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| In [[mathematics]], the simplest '''real analytic Eisenstein series''' is a special function of two variables. It is used in the [[representation theory]] of [[SL2(R)|SL(2,'''R''')]] and in [[analytic number theory]]. It is closely related to the Epstein zeta function.
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| There are many generalizations associated to more complicated groups.
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| ==Definition==
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| The Eisenstein series ''E''(''z'', ''s'') for ''z'' = ''x'' + ''iy'' in the [[upper half-plane]] is defined by
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| :<math>E(z,s) ={1\over 2}\sum_{(m,n)=1}{y^s\over|mz+n|^{2s}}</math>
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| for Re(''s'') > 1, and by analytic continuation for other values of the complex number ''s''. The sum is over all pairs of coprime integers.
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| '''Warning''': there are several other slightly different definitions. Some authors omit the factor of ½, and some sum over all pairs of integers that are not both zero; which changes the function by a factor of ζ(2''s'').
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| ==Properties==
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| ===As a function on ''z''===
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| Viewed as a function of ''z'', ''E''(''z'',''s'') is a real-analytic [[eigenfunction]] of the [[Laplace operator]] on '''H''' with the eigenvalue ''s''(''s''-1). In other words, it satisfies the [[elliptic partial differential equation]]
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| : <math> y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)E(z,s) = s(s-1)E(z,s), </math>    where <math>z=x+yi.</math>
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| The function ''E''(''z'', ''s'') is invariant under the action of SL(2,'''Z''') on ''z'' in the upper half plane by [[fractional linear transformation]]s. Together with the previous property, this means that the Eisenstein series is a [[Maass form]], a real-analytic analogue of a classical elliptic [[modular function]].
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| '''Warning''': ''E''(''z'', ''s'') is not a square-integrable function of ''z'' with respect to the invariant Riemannian metric on '''H'''.
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| ===As a function on ''s''===
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| The Eisenstein series converges for Re(''s'')>1, but can be [[analytic continuation|analytically continued]] to a meromorphic function of ''s'' on the entire complex plane, with a unique pole of residue π at ''s'' = 1 (for all ''z'' in '''H'''). The constant term of the pole at ''s'' = 1 is described by the [[Kronecker limit formula]].
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| The modified function
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| :<math> E^*(z,s) = \pi^{-s}\Gamma(s)\zeta(2s)E(z,s)\ </math>
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| satisfies the functional equation
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| :<math>E^*(z,s) = E^*(z,1-s)\ </math>
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| analogous to the functional equation for the [[Riemann zeta function]] ζ(''s'').
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| Scalar product of two different Eisenstein series ''E''(''z'', ''s'') and ''E''(''z'', ''t'') is given by the [[Maass-Selberg relation]]s.
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| ==Epstein zeta function==
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| The '''Epstein zeta function''' ζ<sub>''Q''</sub>(''s'') {{harv|Epstein|1903}} for a positive definite integral quadratic form ''Q''(''m'', ''n'') = ''cm''<sup>2</sup> + ''bmn'' +''an''<sup>2</sup> is defined by
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| :<math> \zeta_Q(s) = \sum_{(m,n)\ne (0,0)} {1\over Q(m,n)^s}.\ </math>
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| It is essentially a special case of the real analytic Eisenstein series for a special value of ''z'', since
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| :<math> Q(m,n) = a|mz+n|^2\ </math>
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| for
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| :<math> z = {-b\over 2a} + i {\sqrt{4ac-b^2}\over 2a}.\ </math>
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| This zeta function was named after [[Paul Epstein]].
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| ==Generalizations==
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| The real analytic Eisenstein series ''E''(''z'', ''s'') is really the Eisenstein series associated to the discrete subgroup [[modular group|SL(2,'''Z''')]] of [[SL2(R)|SL(2,'''R''')]]. [[Atle Selberg|Selberg]] described generalizations to other discrete subgroups Γ of SL(2,'''R'''), and used these to study the representation of SL(2,'''R''') on L<sup>2</sup>(SL(2,'''R''')/Γ). [[Robert Langlands|Langlands]] extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by [[Joseph Bernstein]].
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| ==See also==
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| *[[Eisenstein series]]
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| *[[Kronecker limit formula]]
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| ==References==
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| *J. Bernstein, [http://www.math.uchicago.edu/~arinkin/langlands/Bernstein/ ''Meromorphic continuation of Eisenstein series'']
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| *{{citation|last=Epstein|first=P.|authorlink=Paul Epstein|title=Zur Theorie allgemeiner Zetafunktionen I|journal=Math. Ann.|volume=56|pages=614–644|year=1903|doi=10.1007/BF01444309|issue=4}}.
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| *{{springer|id=E/e120130|title=Epstein zeta-function|author=A. Krieg}}
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| *{{citation|first=T.|last=Kubota|authorlink=Tomio Kubota|title=Elementary theory of Eisenstein series|isbn=0-470-50920-1|year=1973|publisher=Kodansha|location=Tokyo}}.
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| *{{citation|first=Robert P.|last=Langlands|url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/automorphic.html|title=On the functional equations satisfied by Eisenstein series|isbn=0-387-07872-X|year=1976|publisher=Springer-Verlag|location=Berlin}}.
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| *A. Selberg, ''Discontinuous groups and harmonic analysis'', Proc. Int. Congr. Math., 1962.
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| *[[D. Zagier]], ''Eisenstein series and the Riemann zeta-function''.
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| [[Category:Modular forms]]
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| [[Category:Special functions]]
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| [[Category:Representation theory of Lie groups]]
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| [[Category:Analytic number theory]]
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