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Zeta-function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * Arakawa–Kaneko zeta function * Arithmetic zeta function * Artin–Mazur zeta function of a dynamical system * Barnes zeta function or double zeta function * Beurling zeta function of Beurling generalized primes * Dedekind zeta function of a number field * Duursma zeta function of error-correcting codes * Epstein zeta function of a quadratic form * Goss zeta function of a function field * Hasse–Weil zeta function of a variety * Height zeta function of a variety * Hurwitz zeta function, a generalization of the Riemann zeta function * Igusa zeta function * Ihara zeta function of a graph * ''L''-function, a "twisted" zeta function * Lefschetz zeta function of a morphism * Lerch zeta function, a generalization of the Riemann zeta fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selberg Zeta Function
The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers. If \Gamma is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows, :\zeta_\Gamma(s)=\prod_p(1-N(p)^)^, or :Z_\Gamma(s)=\prod_p\prod^\infty_(1-N(p)^), where ''p'' runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of \Gamma), 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' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > 1 and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lefschetz Zeta Function
In mathematics, the Lefschetz zeta function, zeta-function is a tool used in topological periodic and fixed point (mathematics), fixed point theory, and dynamical systems. Given a continuous map f\colon X\to X, the zeta-function is defined as the formal series :\zeta_f(t) = \exp \left( \sum_^\infty L(f^n) \frac \right), where L(f^n) is the Lefschetz number of the n-th iterated function, iterate of f. This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of f. Examples The identity map on X has Lefschetz zeta function : \frac, where \chi(X) is the Euler characteristic of X, i.e., the Lefschetz number of the identity map. For a less trivial example, let X = S^1 be the unit circle, and let f\colon S^1\to S^1 be reflection in the ''x''-axis, that is, f(\theta) = -\theta. Then f has Lefschetz number 2, while f^2 is the identity map, which has Lefschetz number 0. Likewise, all odd iterates have Le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analytic Eisenstein Series
In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function. There are many generalizations associated to more complicated groups. Definition The Eisenstein series ''E''(''z'', ''s'') for ''z'' = ''x'' + ''iy'' in the upper half-plane is defined by :E(z,s) =\sum_ 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. 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''). Properties As a function on ''z'' 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 wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeta
Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label= Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived from the Phoenician letter zayin . Letters that arose from zeta include the Roman Z and Cyrillic З. Name Unlike the other Greek letters, this letter did not take its name from the Phoenician letter from which it was derived; it was given a new name on the pattern of beta, eta and theta. The word ''zeta'' is the ancestor of ''zed'', the name of the Latin letter Z in Commonwealth English. Swedish and many Romanic languages (such as Italian and Spanish) do not distinguish between the Greek and Roman forms of the letter; "''zeta''" is used to refer to the Roman letter Z as well as the Greek letter. Uses Letter The letter ζ represents the voiced alveolar fricative in Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Gl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lerch Zeta Function
In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887. Definition The Lerch zeta function is given by :L(\lambda, s, \alpha) = \sum_^\infty \frac . A related function, the Lerch transcendent, is given by :\Phi(z, s, \alpha) = \sum_^\infty \frac . The two are related, as :\,\Phi(e^, s,\alpha)=L(\lambda, s, \alpha). Integral representations The Lerch transcendent has an integral representation: : \Phi(z,s,a)=\frac\int_0^\infty \frac\,dt The proof is based on using the integral definition of the Gamma function to write :\Phi(z,s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty t^s z^n e^ \frac and then interchanging the sum and integral. The resulting integral representation converges for z \in \Complex \setm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurwitz Zeta Function
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by :\zeta(s,a) = \sum_^\infty \frac. This series is absolutely convergent for the given values of and and can be extended to a meromorphic function defined for all . The Riemann zeta function is . The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882. Integral representation The Hurwitz zeta function has an integral representation :\zeta(s,a) = \frac \int_0^\infty \frac dx for \operatorname(s)>1 and \operatorname(a)>0. (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing :\zeta(s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty y^s e^ \frac and then interchanging the sum and integral. The integral representation above can be converted to a contour integral representation :\zeta(s,a) = -\Gamma(1-s)\frac \int_C \frac dz where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arakawa–Kaneko Zeta Function
In mathematics, the Arakawa–Kaneko zeta function is a generalisation of the Riemann zeta function which generates special values of the polylogarithm function. Definition The zeta function \xi_k(s) is defined by :\xi_k(s) = \frac \int_0^ \frac\mathrm_k(1-e^) \, dt \ where Li''k'' is the ''k''-th polylogarithm :\mathrm_k(z) = \sum_^ \frac \ . Properties The integral converges for \Re(s) > 0 and \xi_k(s) has analytic continuation to the whole complex plane as an entire function. The special case ''k'' = 1 gives \xi_1(s) = s \zeta(s+1) where \zeta is the Riemann zeta-function. The special case ''s'' = 1 remarkably also gives \xi_k(1) = \zeta(k+1) where \zeta is the Riemann zeta-function. The values at integers are related to multiple zeta function In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by :\zeta(s_1,\ldots,s_k) = \sum_\ \frac = \sum_\ \prod_^k \frac,\! and converge when Re(''s''1) +& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin–Mazur Zeta Function
In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals. It is defined from a given function f as the formal power series :\zeta_f(z)=\exp \left(\sum_^\infty \bigl, \operatorname (f^n)\bigr, \frac \right), where \operatorname (f^n) is the set of fixed points of the nth iterate of the function f, and , \operatorname (f^n), is the number of fixed points (i.e. the cardinality of that set). Note that the zeta function is defined only if the set of fixed points is finite for each n. This definition is formal in that the series does not always have a positive radius of convergence. The Artin–Mazur zeta function is invariant under topological conjugation. The Milnor–Thurston theorem states that the Artin–Mazur zeta function of an interval map f is the inverse of the kneading determinant of f. Analogues The Artin&nd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ihara Zeta Function
In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. The Ihara zeta function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear group. Jean-Pierre Serre suggested in his book ''Trees'' that Ihara's original definition can be reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice in 1985. As observed by Sunada, a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis. Definition The Ihara zeta function is defined as the analytic continuation of the infinite product \zeta_\left(u\right)=\prod_\frac The product in the definition is taken over all prime closed geodesics p of the graph G = (V, E), where geodesics which differ by a cyclic rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matsumoto Zeta Function
In mathematics, Matsumoto zeta functions are a type of zeta function introduced by Kohji Matsumoto in 1990. They are functions of the form :\phi(s)=\prod_\frac where ''p'' is a prime and ''A''''p'' is a polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa .... References * Zeta and L-functions {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Zeta Function
In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by :\zeta(s_1,\ldots,s_k) = \sum_\ \frac = \sum_\ \prod_^k \frac,\! and converge when Re(''s''1) + ... + Re(''s''''i'') > ''i'' for all ''i''. Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example, Zhao (1999)). When ''s''1, ..., ''s''''k'' are all positive integers (with ''s''1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums. These values can also be regarded as special values of the multiple polylogarithms. The ''k'' in the above definition is named the "depth" of a MZV, and the ''n'' = ''s''1 + ... + ''s''''k'' is known as the "weight". The standard shorthand for writing multiple zeta functions is to place repeating strings of the argument within braces and use a superscript to indicate the nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
