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mathematical analysis > functional analysis > spectral theory > Selberg zeta function

algebra > linear algebra > spectral theory > Selberg zeta function

number > number theory > analytic number theory > L-function > Selberg zeta function

mathematical analysis > complex analysis > meromorphic function > L-function > Selberg zeta function

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Selberg zeta function

The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function
${\\displaystyle \\zeta (s)=\\prod _{p\\in \\mathbb {P} }{\ rac {1}{1-p^{-s}}}}$
where ${\\displaystyle \\mathbb {P} }$ is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the prime numbers. If ${\\displaystyle \\Gamma }$ is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows,
${\\displaystyle \\zeta _{\\Gamma }(s)=\\prod _{p}(1-N(p)^{-s})^{-1},}$
or
${\\displaystyle Z_{\\Gamma }(s)=\\prod _{p}\\prod _{n=0}^{\\infty }(1-N(p)^{-s-n}),}$
where p runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of ${\\displaystyle \\Gamma }$ ), and N(p) denotes the length of p (equivalently, the square of the bigger eigenvalue of p).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Selberg_zeta_function)

fonction zêta de Selberg

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