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The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function
where is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the prime numbers. If is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows,
or
where p runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of ), and N(p) denotes the length of p (equivalently, the square of the bigger eigenvalue of p).
For any 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 geodesics 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 there exists a zero at the point . 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, . The order of the zero equals the order of the corresponding pole of the scattering matrix.
The zeta-function also has poles at , and can have zeros or poles at the points .
The Ihara zeta function is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function.
and 19 Related for: Selberg zeta function information
The Selbergzeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zetafunction ζ ( s ) = ∏ p ∈ P 1 1 − p − s {\displaystyle...
characteristic zero. Selberg (1956) introduced the Selbergzetafunction of a Riemann surface. These are similar to the Riemann zetafunction: they have a functional...
by the analogy, Selberg introduced the Selbergzetafunction of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula....
mathematics, the Ihara zetafunction is a zetafunction associated with a finite graph. It closely resembles the Selbergzetafunction, and is used to relate...
numbers and the zeros of the zetafunction. He generally worked alone. His only coauthor was Sarvadaman Chowla. Selberg was awarded the 1986 Wolf Prize...
universality of zetafunctions is the remarkable ability of the Riemann zetafunction and other similar functions (such as the Dirichlet L-functions) to approximate...
The Riemann zetafunction or Euler–Riemann zetafunction, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...
In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms...
for the Riemann zetafunction ζ(s). Scalar product of two different Eisenstein series E(z, s) and E(z, t) is given by the Maass-Selberg relations. The...
_{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}},} which is a special case of the Rankin–Selberg convolution. A Lambert...
theorem for the gamma functions can be understood to be a special case, for the trivial Dirichlet character, of the Chowla–Selberg formula. Formally similar...
discontinuous groups (Fuchsian groups), different zetafunctions (for example those of Ruelle and Selberg, in particular those associated with certain groups...
are improved upon. Also, there is a Selbergzetafunction which is formally similar to the usual Riemann zetafunction and shares many of its properties...
An arithmetical function f {\displaystyle f} is Selberg multiplicative if for each prime p {\displaystyle p} there exists a function f p {\displaystyle...
particular, the Riemann zetafunction). The first such distribution found is π(N) ~ N/log(N), where π(N) is the prime-counting function (the number of primes...
formula, via the Selbergzetafunction. Sunada noticed that the method of constructing number fields with the same Dedekind zetafunction could be adapted...
Horn: Horn hypergeometric series Adolf Hurwitz: Hurwitz zeta-function Hypergeometric function 2F1 Henry Jack (1917–1978) Dundee: Jack polynomial F. H...
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