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Shintani Zeta Function
In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by . They include Hurwitz zeta functions and Barnes zeta functions. Definition Let P(\mathbf) be a polynomial in the variables \mathbf=(x_1,\dots,x_r) with real coefficients such that P(\mathbf) is a product of linear polynomials with positive coefficients, that is, P(\mathbf)=P_1(\mathbf)P_2(\mathbf)\cdots P_k(\mathbf), where P_i(\mathbf)= a_ x_1 + a_ x_2 +\cdots + a_x_r + b_i, where a_>0, b_i>0 and k=\deg P. The ''Shintani zeta function'' in the variable s is given by (the meromorphic continuation of)\zeta(P;s)=\sum_^\frac. The multi-variable version The definition of Shintani zeta function has a straightforward generalization to a zeta function in several variables (s_1,\dots,s_k) given by\sum_^\frac.The special case when ''k'' = 1 is the Barnes zeta function In mathematics, a Barnes zeta function is a generalization of the Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
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] |
Barnes Zeta Function
In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by . It is further generalized by the Shintani zeta function. Definition The Barnes zeta function is defined by : \zeta_N(s,w\mid a_1,\ldots,a_N)=\sum_\frac where ''w'' and ''a''''j'' have positive real part and ''s'' has real part greater than ''N''. It has a meromorphic continuation to all complex ''s'', whose only singularities are simple poles at ''s'' = 1, 2, ..., ''N''. For ''N'' = ''w'' = ''a''1 = 1 it is the Riemann zeta function. References * * * * * Zeta and L-functions {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witten Zeta Function
In mathematics, the Witten zeta function, is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio .... These zeta functions were introduced by Don Zagier who named them after Edward Witten's study of their special values (among other things). Note that in, Witten zeta functions do not appear as explicit objects in their own right. Definition If G is a compact semisimple Lie group, the associated Witten zeta function is (the meromorphic continuation of) the series :\zeta_G(s)=\sum_\rho\frac, where the sum is over equivalence classes of irreducible representations of G. In the case where G is connected and simply connected, the correspondence between representa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
