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Chowla–Selberg Formula
In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational numbers. The result was essentially found by and rediscovered by . Statement In logarithmic form, the Chowla–Selberg formula states that in certain cases the sum : \frac\sum_r \chi(r)\log \Gamma\left( \frac \right) = \frac\log(4\pi\sqrt) +\sum_\tau\log\left(\sqrt, \eta(\tau), ^2\right) can be evaluated using the Kronecker limit formula. Here χ is the quadratic residue symbol modulo ''D'', where ''−D'' is the discriminant of an imaginary quadratic field. The sum is taken over 0 < ''r'' < ''D'', with the usual convention χ(''r'') = 0 if ''r'' and ''D'' have a common factor. The function η is the Dedekind eta function, and ''h'' is the class number, and ''w'' is the number of roots of unity. Origin and applications The origin of such formu ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Gamma Function
In mathematics, the ''p''-adic gamma function Γ''p'' is a function of a ''p''-adic variable analogous to the gamma function. It was first explicitly defined by , though pointed out that implicitly used the same function. defined a ''p''-adic analog ''G''''p'' of log Γ. had previously given a definition of a different ''p''-adic analogue of the gamma function, but his function does not have satisfactory properties and is not used much. Definition The ''p''-adic gamma function is the unique continuous function of a ''p''-adic integer ''x'' (with values in \mathbb_p) such that :\Gamma_p(x) = (-1)^x \prod_ i for positive integers ''x'', where the product is restricted to integers ''i'' not divisible by ''p''. As the positive integers are dense with respect to the ''p''-adic topology in \mathbb_p, \Gamma_p(x) can be extended uniquely to the whole of \mathbb_p. Here \mathbb_p is the ring of ''p''-adic integers. It follows from the definition that the values of \Gamma_p( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Des Sciences Mathématiques
Bulletin or The Bulletin may refer to: Periodicals (newspapers, magazines, journals) * ''Bulletin'' (online newspaper), a Swedish online newspaper * ''The Bulletin'' (Australian periodical), an Australian magazine (1880–2008) ** Bulletin Debate, a famous dispute from 1892 to 1893 between Henry Lawson and Banjo Paterson * ''The Bulletin'' (alternative weekly), an alternative weekly published in Montgomery County, Texas, U.S. * ''The Bulletin'' (Bend), a daily newspaper in Bend, Oregon, U.S. * ''The Bulletin'' (Belgian magazine), a weekly English-language magazine published in Brussels, Belgium * ''The Bulletin'' (Philadelphia newspaper), a newspaper in Philadelphia, Pennsylvania, U.S. (2004–2009) * ''The Bulletin'' (Norwich) * ''London Bulletin'', surrealist monthly magazine (1938–1940) * ''The Morning Bulletin'', a daily newspaper published in Rockhampton, Queensland, Australia since 1861 * ''Philadelphia Bulletin'', a newspaper published in Philadelphia, U.S. (1847� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Für Die Reine Und Angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Daniel Huybrechts (Rheinische Friedrich-Wilhelms-Universität Bonn). Past editors * 1826–1856: August Leopold Crelle * 1856–1880: Carl Wilhelm Borchardt * 1881–1888: Leopold Kronecker, Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The National Academy Of Sciences
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2022 impact factor of 9.4. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the past, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open-access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are available ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication Theorem
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises. Finite characteristic The multiplication theorem takes two common forms. In the first case, a finite number of terms are added or multiplied to give the relation. In the second case, an infinite number of terms are added or multiplied. The finite form typically occurs only for the gamma and related functions, for which the identity follows from a p-adic relation over a finite field. For example, the multiplication theorem for the gamma function follows from the Chowla–Selberg formula, which follows from the theory of comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Reflection Formula
In mathematics, a reflection formula or reflection relation for a function is a relationship between and . It is a special case of a functional equation. It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae. Reflection formulae are useful for numerical computation of special functions. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the complex plane) can be employed for all arguments. Known formulae The even and odd functions satisfy by definition simple reflection relations around . For all even functions, f(-x) = f(x), and for all odd functions, f(-x) = -f(x). A famous relationship is Euler's reflection formula \Gamma(z)\Gamma(1-z) = \frac, \qquad z \not\in \mathbb Z for the gamma function \Gamma(z), due to Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gross–Koblitz Formula
In mathematics, the Gross–Koblitz formula, introduced by expresses a Gauss sum using a product of values of the ''p''-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation and generalizes the Stickelberger theorem. gave another proof of the Gross–Koblitz formula ("Boyarsky" being a pseudonym of Bernard Dwork Bernard Morris Dwork (May 27, 1923 – May 9, 1998) was an American mathematician, known for his application of ''p''-adic analysis to local zeta functions, and in particular for a proof of the first part of the Weil conjectures: the rationality ...), and gave an elementary proof. Statement The Gross–Koblitz formula states that the Gauss sum \tau can be given in terms of the p-adic gamma function \Gamma_p by :\tau_q(r) = -\pi^\prod_\Gamma_p\!\left(\frac \right) where * q is a power p^f of a prime p, * r is an integer with 0 \leq r < q-1, * is the inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety Of CM-type
In mathematics, an abelian variety ''A'' defined over a field ''K'' is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(''A''). The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century. One of the major achievements in algebraic number theory and algebraic geometry of the twentieth century was to find the correct formulations of the corresponding theory for abelian varieties of dimension ''d'' > 1. The problem is at a deeper level of abstraction, because it is much harder to manipulate analytic functions of several complex variables. The formal definition is that : \operatorname_\mathbb(A) the tensor product of End(''A'') with the rational number field Q, should contain a commutative subring of dimension 2''d'' over Q. When ''d'' = 1 this can only be a quadratic field, and one recovers the cases where End(''A'') is an order in an imaginary quadratic field. For ''d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
