Special functions
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Analytic number theory
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Algebraic number theory
Zeta and L-functions
Researchers announce new way to explore mathematical universe
An international group of mathematicians at Arizona State University and other institutions have released a new kind of online resource to help discover uncharted mathematical worlds.
Researchers announce new way to explore mathematical universe
An international group of mathematicians at Arizona State University and other institutions have released a new kind of online resource to help discover uncharted mathematical worlds.
Theta correspondence
In mathematics, the theta correspondence or Howe correspondence is a correspondence between automorphic forms associated to the two groups of a dual reductive pair, introduced by Howe (1979).
Dirichlet beta function
In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the ...
Li's criterion
In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is completely equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li...
Rankin–Selberg method
In mathematics, the Rankin–Selberg method, introduced by (Rankin 1939) and Selberg (1940), also known as the theory of integral representations of L-functions, is a technique for directly co...
Explicit formula
Explicit formula can refer to:
Weil's criterion
In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized ...
Barnes zeta function
In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by E. W. Barnes (1901). It is further generalized by the Shintani zeta function.
The Bar...
Selberg class
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 that seem to capture the essential propert...
Brumer–Stark conjecture
Weil conjectures
In mathematics, the Weil conjectures were some highly influential proposals by André Weil (1949) on the generating functions (known as local zeta-functions) derived from counting the number ...
Ramanujan–Petersson conjecture
In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p.176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of ...
Shimura variety
In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group...
Lefschetz zeta function
In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a mapping f, the zeta-function is defined as the formal serieswh...
Local Langlands conjectures
In mathematics, the local Langlands conjectures, introduced by Langlands (1967, 1970), are part of the Langlands program. They describe a correspondence between the complex representations of a ...
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2. The name is also ...
Riemann hypothesis - Wikipedia
Multiple zeta function
In mathematics, the multiple zeta functions are generalisations of the Riemann zeta function, defined byand converge when Re(s1) + ... + Re(si) > i for all i. Lik...
Apéry's theorem
In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the numbercannot be written as a fraction p/q with p and q being integers.Th...
Apéry's theorem - Wikipedia
Apéry's constant
In mathematics, Apéry's constant is a number that occurs in a variety of situations. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electro...
Apéry's constant - Wikipedia
Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1734 and read on 5 December 1735 in The S...
Riemann–Siegel theta function
In mathematics, the Riemann–Siegel theta function is defined in terms of the Gamma function asfor real values of t. Here the argument is chosen in such a way that a continuous function is obtained an...
Riemann–Siegel theta function - Wikipedia
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 Takuro Shintani (1976). They include Hurwitz zeta ...
Langlands program
In mathematics, the Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algeb...
Selberg's zeta function conjecture
In mathematics, the Selberg conjecture, named after Atle Selberg, is about the density of zeros of the Riemann zeta function ζ(1/2 + it). It is known that the function has infinitely ma...
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. It is widely recognized as one of the most challenging mathematical problems; the conjecture ...
Functional equation (L-function)
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of...
Minakshisundaram–Pleijel zeta function
The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introducedby Subbaramiah Minakshisundaram and&...
Minakshisundaram–Pleijel zeta function - Wikipedia
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 ...
Automorphic L-function
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic form π of a reductive group G over a global field and a finite-dimensional comple...