71.234.3.238: /* Partial derivative */

2013-10-27T23:11:43Z

Partial derivative

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	~~Hi there~~, ~~I am Sophia~~. For a ~~whilst I~~'~~ve been~~ in ~~Mississippi but now I'm contemplating other options~~. ~~My working day occupation~~ is an ~~info officer but I've already applied~~ for ~~an additional 1~~. ~~To climb~~ is ~~some thing she would~~ by ~~no means give up~~.<br><br>~~my web site: free psychic reading~~ ([~~http:~~/~~/203~~.~~250~~.78.~~160~~/~~zbxe/?document_srl~~=~~1792908 just click the next website page~~])		The '''Selberg zeta-function''' was introduced by {{harvs\|txt\|authorlink=Atle Selberg\|first=Atle \|last=Selberg\|year=1956}}. It is analogous to the famous [[Riemann zeta function]]
			:<math> \zeta(s) = \prod_{p\in\mathbb{P}} \frac{1}{1-p^{-s}} </math>
			where <math> \mathbb{P} </math> is the set of prime numbers. The Selberg zeta-function uses the lengths of simple [[closed geodesic]]s instead of the primes numbers.　If <math>\Gamma</math> is a subgroup of SL(2,'''R''') Selberg zeta function is defined as follows,
			:<math>\zeta_\Gamma(s)=\prod_p(1-N(p)^{-s})^{-1},</math>
			or
			:<math>Z_\Gamma(s)=\prod_p\prod^\infty_{n=0}(1-N(p)^{-s-n}),</math>
			where p run all over the prime congruent class and N(p) is the norm of congruent class p, which is square of the bigger eigenvalue of p.

			For any [[hyperbolic manifold \| hyperbolic surface]] of finite area there is an associated '''Selberg zeta-function'''; this function is a [[meromorphic function]] defined in the [[complex plane]]. The zeta function is defined in terms of the closed [[geodesic]]s of the surface.

			The zeros and poles of the Selberg zeta-function, ''Z''(''s''), can be described in terms of spectral data of the surface.

			The zeros are at the following points:
			# For every cusp form with eigenvalue <math>s_0(1-s_0)</math> there exists a zero at the point <math>s_0</math>. The order of the zero equals the dimension of the corresponding eigenspace. (A cusp form is an eigenfunction to the [[Laplace-Beltrami operator]] which has [[Fourier expansion]] with zero constant term.)
			# The zeta-function also has a zero at every pole of the determinant of the scattering matrix, <math> \phi(s) </math>. The order of the zero equals the order of the corresponding pole of the scattering matrix.

			The zeta-function also has poles at <math> 1/2 - \mathbb{N} </math>, and can have zeros or poles at the points <math> - \mathbb{N} </math>.

			== Selberg zeta-function for the modular group ==
			For the case where the surface is <math> \Gamma \backslash \mathbb{H}^2 </math>, where <math> \Gamma </math> is the [[modular group]], the Selberg zeta-function is of special interest. For this special case the Selberg zeta-function is intimately connected to the [[Riemann zeta function\|Riemann zeta-function]].

			In this case the determinant of the [[scattering matrix]] is given by:
			:<math> \varphi(s) = \pi^{1/2} \frac{ \Gamma(s-1/2) \zeta(2s-1) }{ \Gamma(s) \zeta(2s) }. </math>

			In particular, we see that if the Riemann zeta-function has a zero at <math>s_0</math>, then the determinant of the scattering matrix has a pole at <math>s_0/2</math>, and hence the Selberg zeta-function has a zero at <math>s_0/2</math>.

			==References==
			*{{Citation \| last1=Fischer \| first1=Jürgen \| title=An approach to the Selberg trace formula via the Selberg zeta-function \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Mathematics \| isbn=978-3-540-15208-8 \| doi=10.1007/BFb0077696 \| id={{MathSciNet \| id = 892317}} \| year=1987 \| volume=1253}}
			*{{Citation \| last1=Hejhal \| first1=Dennis A. \| title=The Selberg trace formula for PSL(2,R). Vol. I \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Mathematics, Vol. 548 \| doi=10.1007/BFb0079608 \| id={{MathSciNet \| id = 0439755}} \| year=1976 \| volume=548}}
			*{{Citation \| last1=Hejhal \| first1=Dennis A. \| title=The Selberg trace formula for PSL(2,R). Vol. 2 \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Lecture Notes in Mathematics \| isbn=978-3-540-12323-1 \| doi=10.1007/BFb0061302 \| id={{MathSciNet \| id = 711197}} \| year=1983 \| volume=1001}}
			* [[Henryk Iwaniec\|Iwaniec, H.]] Spectral methods of automorphic forms, American Mathematical Society, second edition, 2002.
			*{{Citation \| last1=Selberg \| first1=Atle \| title=Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series \| id={{MathSciNet \| id = 0088511}} \| year=1956 \| journal=J. Indian Math. Soc. (N.S.) \| volume=20 \| pages=47–87}}
			* Venkov, A. B. Spectral theory of automorphic functions. Proc. Steklov. Inst. Math, 1982.

			[[Category:Zeta and L-functions]]
			[[Category:Spectral theory]]

en>Download: gen fixes, typos fixed: a a → a using AWB

2012-05-06T21:06:37Z

gen fixes, typos fixed: a a → a using AWB

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Hi there, I am Sophia. For a whilst I've been in Mississippi but now I'm contemplating other options. My working day occupation is an info officer but I've already applied for an additional 1. To climb is some thing she would by no means give up.<br><br>my web site: free psychic reading ([http://203.250.78.160/zbxe/?document_srl=1792908 just click the next website page])

Vector-valued function - Revision history

71.234.3.238: /* Partial derivative */

en>Download: gen fixes, typos fixed: a a → a using AWB