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Riemann zeta function, the Glossary

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).^[1]

151 relations: Abel–Plana formula, Abramowitz and Stegun, Absolute convergence, Algebraic number theory, American Mathematical Society, Analytic continuation, Analytic function, Analytic number theory, Anatoly Karatsuba, Apéry's constant, Argument (complex analysis), Arithmetic zeta function, Atle Selberg, Basel problem, Bernhard Riemann, Bernoulli number, Brady Haran, Canadian Mathematical Society, Casimir effect, Cauchy principal value, Chebyshev polynomials, Clausen function, Clay Mathematics Institute, Complex analysis, Complex number, Complex plane, Comptes rendus de l'Académie des Sciences, Conjecture, Contour integration, Convergent series, Coprime integers, Dedekind zeta function, Dirichlet eta function, Dirichlet L-function, Dirichlet series, Divergent series, Dynamical system, Engel expansion, Equal temperament, Eric W. Weisstein, Essential singularity, Euclid's theorem, Euler product, Euler summation, Euler's constant, Euler–Maclaurin formula, Falling and rising factorials, Function (mathematics), Functional equation, Fundamental theorem of arithmetic, ... Expand index (101 more) »
Bernhard Riemann

Abel–Plana formula

In mathematics, the Abel–Plana formula is a summation formula discovered independently by and.

See Riemann zeta function and Abel–Plana formula

Abramowitz and Stegun (AS) is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the National Institute of Standards and Technology (NIST).

See Riemann zeta function and Abramowitz and Stegun

Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

See Riemann zeta function and Absolute convergence

Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.

See Riemann zeta function and Algebraic number theory

American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

See Riemann zeta function and American Mathematical Society

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function.

See Riemann zeta function and Analytic continuation

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

See Riemann zeta function and Analytic function

Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.

See Riemann zeta function and Analytic number theory

Anatoly Karatsuba

Anatoly Alexeyevich Karatsuba (his first name often spelled Anatolii) (Анато́лий Алексе́евич Карацу́ба; Grozny, Soviet Union, 31 January 1937 – Moscow, Russia, 28 September 2008) was a Russian mathematician working in the field of analytic number theory, ''p''-adic numbers and Dirichlet series.

See Riemann zeta function and Anatoly Karatsuba

Apéry's constant

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. Riemann zeta function and Apéry's constant are analytic number theory and zeta and L-functions.

See Riemann zeta function and Apéry's constant

Argument (complex analysis)

In mathematics (particularly in complex analysis), the argument of a complex number, denoted, is the angle between the positive real axis and the line joining the origin and, represented as a point in the complex plane, shown as \varphi in Figure 1.

See Riemann zeta function and Argument (complex analysis)

Arithmetic zeta function

In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. Riemann zeta function and arithmetic zeta function are zeta and L-functions.

See Riemann zeta function and Arithmetic zeta function

Atle Selberg

Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory.

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Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. Riemann zeta function and Basel problem are zeta and L-functions.

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Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry.

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Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis.

See Riemann zeta function and Bernoulli number

Brady Haran

Brady John Haran (born 18 June 1976) is an Australian-British independent filmmaker and video journalist who produces educational videos and documentary films for his YouTube channels, the most notable being Computerphile and Numberphile.

See Riemann zeta function and Brady Haran

Canadian Mathematical Society

The Canadian Mathematical Society (CMS; Société mathématique du Canada) is an association of professional mathematicians dedicated to the interests of mathematical research, outreach, scholarship and education in Canada.

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Casimir effect

In quantum field theory, the Casimir effect (or Casimir force) is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field.

See Riemann zeta function and Casimir effect

Cauchy principal value

In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.

See Riemann zeta function and Cauchy principal value

Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x).

See Riemann zeta function and Chebyshev polynomials

Clausen function

In mathematics, the Clausen function, introduced by, is a transcendental, special function of a single variable. Riemann zeta function and Clausen function are zeta and L-functions.

See Riemann zeta function and Clausen function

Clay Mathematics Institute

The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge.

See Riemann zeta function and Clay Mathematics Institute

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

See Riemann zeta function and Complex analysis

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

See Riemann zeta function and Complex number

Complex plane

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers.

See Riemann zeta function and Complex plane

Comptes rendus de l'Académie des Sciences

(English: Proceedings of the Academy of Sciences), or simply Comptes rendus, is a French scientific journal published since 1835.

See Riemann zeta function and Comptes rendus de l'Académie des Sciences

Conjecture

In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof.

See Riemann zeta function and Conjecture

Contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

See Riemann zeta function and Contour integration

Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

See Riemann zeta function and Convergent series

Coprime integers

In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.

See Riemann zeta function and Coprime integers

Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field of rational numbers Q). Riemann zeta function and Dedekind zeta function are zeta and L-functions.

See Riemann zeta function and Dedekind zeta function

Dirichlet eta function

In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s). Riemann zeta function and Dirichlet eta function are zeta and L-functions.

See Riemann zeta function and Dirichlet eta function

Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form where \chi is a Dirichlet character and s a complex variable with real part greater than 1. Riemann zeta function and Dirichlet L-function are zeta and L-functions.

See Riemann zeta function and Dirichlet L-function

Dirichlet series

In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where s is complex, and a_n is a complex sequence. Riemann zeta function and Dirichlet series are zeta and L-functions.

See Riemann zeta function and Dirichlet series

Divergent series

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

See Riemann zeta function and Divergent series

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve.

See Riemann zeta function and Dynamical system

Engel expansion

The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers (a_1,a_2,a_3,\dots) such that For instance, Euler's number ''e'' has the Engel expansion corresponding to the infinite series Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion.

See Riemann zeta function and Engel expansion

Equal temperament

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same.

See Riemann zeta function and Equal temperament

Eric W. Weisstein

Eric Wolfgang Weisstein (born March 18, 1969) is an American scientist, mathematician, and encyclopedist who created and maintains the encyclopedias MathWorld and ScienceWorld.

See Riemann zeta function and Eric W. Weisstein

Essential singularity

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior.

See Riemann zeta function and Essential singularity

Euclid's theorem

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers.

See Riemann zeta function and Euclid's theorem

Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. Riemann zeta function and Euler product are analytic number theory and zeta and L-functions.

See Riemann zeta function and Euler product

Euler summation

In the mathematics of convergent and divergent series, Euler summation is a summation method.

See Riemann zeta function and Euler summation

Euler's constant

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma, defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by: \begin \gamma &.

See Riemann zeta function and Euler's constant

Euler–Maclaurin formula

In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum.

See Riemann zeta function and Euler–Maclaurin formula

Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n.

See Riemann zeta function and Falling and rising factorials

Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See Riemann zeta function and Function (mathematics)

Functional equation

In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns.

See Riemann zeta function and Functional equation

Fundamental theorem of arithmetic

In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors.

See Riemann zeta function and Fundamental theorem of arithmetic

G. H. Hardy

Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis.

See Riemann zeta function and G. H. Hardy

Gamma function

In mathematics, the gamma function (represented by, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers.

See Riemann zeta function and Gamma function

Gauss–Kuzmin–Wirsing operator

In mathematics, the Gauss–Kuzmin–Wirsing operator is the transfer operator of the Gauss map that takes a positive number to the fractional part of its reciprocal.

See Riemann zeta function and Gauss–Kuzmin–Wirsing operator

Generalized Riemann hypothesis

The Riemann hypothesis is one of the most important conjectures in mathematics. Riemann zeta function and Generalized Riemann hypothesis are Bernhard Riemann and zeta and L-functions.

See Riemann zeta function and Generalized Riemann hypothesis

Geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms.

See Riemann zeta function and Geometric series

Graduate Texts in Mathematics

Graduate Texts in Mathematics (GTM) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.

See Riemann zeta function and Graduate Texts in Mathematics

Greek alphabet

The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC.

See Riemann zeta function and Greek alphabet

Hankel contour

In mathematics, a Hankel contour is a path in the complex plane which extends from (+∞,δ), around the origin counter clockwise and back to (+∞,−δ), where δ is an arbitrarily small positive number.

See Riemann zeta function and Hankel contour

Harmonic series (mathematics)

In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_^\infty\frac.

See Riemann zeta function and Harmonic series (mathematics)

Harmonic series (music)

A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental frequency.

See Riemann zeta function and Harmonic series (music)

Helmut Hasse

Helmut Hasse (25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and diophantine geometry (Hasse principle), and to local zeta functions.

See Riemann zeta function and Helmut Hasse

Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space.

See Riemann zeta function and Holomorphic function

Hurwitz zeta function

In mathematics, the Hurwitz zeta function is one of the many zeta functions. Riemann zeta function and Hurwitz zeta function are zeta and L-functions.

See Riemann zeta function and Hurwitz zeta function

Infinite product

In mathematics, for a sequence of complex numbers a1, a2, a3,...

See Riemann zeta function and Infinite product

Infinity

Infinity is something which is boundless, endless, or larger than any natural number.

See Riemann zeta function and Infinity

International Journal of Mathematics and Mathematical Sciences

The International Journal of Mathematics and Mathematical Sciences is a biweekly peer-reviewed mathematics journal.

See Riemann zeta function and International Journal of Mathematics and Mathematical Sciences

Jacques Hadamard

Jacques Salomon Hadamard (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations.

See Riemann zeta function and Jacques Hadamard

John Edensor Littlewood

John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician.

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John Tate (mathematician)

John Torrence Tate Jr. (March 13, 1925 – October 16, 2019) was an American mathematician distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry, and related areas in algebraic geometry.

See Riemann zeta function and John Tate (mathematician)

Jordan's totient function

In number theory, Jordan's totient function, denoted as J_k(n), where k is a positive integer, is a function of a positive integer, n, that equals the number of k-tuples of positive integers that are less than or equal to n and that together with n form a coprime set of k+1 integers Jordan's totient function is a generalization of Euler's totient function, which is the same as J_1(n).

See Riemann zeta function and Jordan's totient function

Joseph Ser

Joseph Ser (1875–1954) was a French mathematician, of whom little was known till now.

See Riemann zeta function and Joseph Ser

Journal of Computational and Applied Mathematics

The Journal of Computational and Applied Mathematics is a peer-reviewed scientific journal covering computational and applied mathematics.

See Riemann zeta function and Journal of Computational and Applied Mathematics

Journal of Number Theory

The Journal of Number Theory (JNT) is a monthly peer-reviewed scientific journal covering all aspects of number theory.

See Riemann zeta function and Journal of Number Theory

Kanakanahalli Ramachandra

Kanakanahalli Ramachandra (18 August 1933 – 17 January 2011) was an Indian mathematician working in both analytic number theory and algebraic number theory.

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Konrad Knopp

Konrad Hermann Theodor Knopp (22 July 1882 – 20 April 1957) was a German mathematician who worked on generalized limits and complex functions.

See Riemann zeta function and Konrad Knopp

L-function

In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. Riemann zeta function and l-function are zeta and L-functions.

See Riemann zeta function and L-function

Lambert W function

In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function, where is any complex number and is the exponential function.

See Riemann zeta function and Lambert W function

Laurent series

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.

See Riemann zeta function and Laurent series

Lehmer pair

In the study of the Riemann hypothesis, a Lehmer pair is a pair of zeros of the Riemann zeta function that are unusually close to each other. Riemann zeta function and Lehmer pair are analytic number theory.

See Riemann zeta function and Lehmer pair

Leonhard Euler

Leonhard Euler (15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus.

See Riemann zeta function and Leonhard Euler

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. Riemann zeta function and Lerch zeta function are zeta and L-functions.

See Riemann zeta function and Lerch zeta function

Limit of a sequence

As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1.

See Riemann zeta function and Limit of a sequence

List of zeta functions

In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function Zeta functions include. Riemann zeta function and List of zeta functions are zeta and L-functions.

See Riemann zeta function and List of zeta functions

Mathematische Zeitschrift

Mathematische Zeitschrift (German for Mathematical Journal) is a mathematical journal for pure and applied mathematics published by Springer Verlag.

See Riemann zeta function and Mathematische Zeitschrift

Möbius function

The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832.

See Riemann zeta function and Möbius function

Möbius inversion formula

In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors.

See Riemann zeta function and Möbius inversion formula

Mean value theorem

In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

See Riemann zeta function and Mean value theorem

Mellin transform

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.

See Riemann zeta function and Mellin transform

Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function.

See Riemann zeta function and Meromorphic function

Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane, \,\mathcal\,, that satisfies. Riemann zeta function and modular form are analytic number theory.

See Riemann zeta function and Modular form

Multiple zeta function

In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by and converge when Re(s1) +... + Re(si) > i for all i. Riemann zeta function and multiple zeta function are zeta and L-functions.

See Riemann zeta function and Multiple zeta function

Multiplicative function

In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1).

See Riemann zeta function and Multiplicative function

Musical tuning

In music, there are two common meanings for tuning.

See Riemann zeta function and Musical tuning

Natural density

In number theory, natural density, also referred to as asymptotic density or arithmetic density, is one method to measure how "large" a subset of the set of natural numbers is.

See Riemann zeta function and Natural density

Odlyzko–Schönhage algorithm

In mathematics, the Odlyzko–Schönhage algorithm is a fast algorithm for evaluating the Riemann zeta function at many points, introduced by. Riemann zeta function and Odlyzko–Schönhage algorithm are analytic number theory and zeta and L-functions.

See Riemann zeta function and Odlyzko–Schönhage algorithm

On the Number of Primes Less Than a Given Magnitude

" die Anzahl der Primzahlen unter einer gegebenen " (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin. Riemann zeta function and on the Number of Primes Less Than a Given Magnitude are analytic number theory and Bernhard Riemann.

See Riemann zeta function and On the Number of Primes Less Than a Given Magnitude

Oxford

Oxford is a city and non-metropolitan district in Oxfordshire, England, of which it is the county town.

See Riemann zeta function and Oxford

Pafnuty Chebyshev

Pafnuty Lvovich Chebyshev (p) (–) was a Russian mathematician and considered to be the founding father of Russian mathematics.

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Peter Borwein

Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953 – 23 August 2020) was a Canadian mathematician and a professor at Simon Fraser University.

See Riemann zeta function and Peter Borwein

Physics

Physics is the natural science of matter, involving the study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.

See Riemann zeta function and Physics

Poisson summation formula

In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform.

See Riemann zeta function and Poisson summation formula

Polygamma function

In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: Thus holds where is the digamma function and is the gamma function.

See Riemann zeta function and Polygamma function

Polylogarithm

In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument. Riemann zeta function and polylogarithm are zeta and L-functions.

See Riemann zeta function and Polylogarithm

Power law

In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to a power of the change, independent of the initial size of those quantities: one quantity varies as a power of another.

See Riemann zeta function and Power law

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

See Riemann zeta function and Prime number

Prime number theorem

In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers.

See Riemann zeta function and Prime number theorem

Prime zeta function

In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by. Riemann zeta function and prime zeta function are zeta and L-functions.

See Riemann zeta function and Prime zeta function

Prime-counting function

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number. Riemann zeta function and prime-counting function are analytic number theory.

See Riemann zeta function and Prime-counting function

Primorial

In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.

See Riemann zeta function and Primorial

Probability theory

Probability theory or probability calculus is the branch of mathematics concerned with probability.

See Riemann zeta function and Probability theory

Proceedings of the American Mathematical Society

Proceedings of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.

See Riemann zeta function and Proceedings of the American Mathematical Society

Proof of the Euler product formula for the Riemann zeta function

Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations about Infinite Series), published by St Petersburg Academy in 1737. Riemann zeta function and Proof of the Euler product formula for the Riemann zeta function are zeta and L-functions.

See Riemann zeta function and Proof of the Euler product formula for the Riemann zeta function

Pure mathematics

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics.

See Riemann zeta function and Pure mathematics

Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics.

See Riemann zeta function and Quantum field theory

Ramanujan summation

Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.

See Riemann zeta function and Ramanujan summation

Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

See Riemann zeta function and Rational number

Rational zeta series

In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Riemann zeta function and rational zeta series are zeta and L-functions.

See Riemann zeta function and Rational zeta series

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

See Riemann zeta function and Real number

Regularization (physics)

In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter called the regulator.

See Riemann zeta function and Regularization (physics)

Renormalization

Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions.

See Riemann zeta function and Renormalization

Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.

See Riemann zeta function and Residue (complex analysis)

Riemann hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part. Riemann zeta function and Riemann hypothesis are analytic number theory, Bernhard Riemann and zeta and L-functions.

See Riemann zeta function and Riemann hypothesis

Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. Riemann zeta function and Riemann sphere are Bernhard Riemann.

See Riemann zeta function and Riemann sphere

Riemann Xi function

In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. Riemann zeta function and Riemann Xi function are Bernhard Riemann and zeta and L-functions.

See Riemann zeta function and Riemann Xi function

Riemann–Siegel theta function

In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as for real values of t. Riemann zeta function and Riemann–Siegel theta function are Bernhard Riemann and zeta and L-functions.

See Riemann zeta function and Riemann–Siegel theta function

Roger Apéry

Roger Apéry (14 November 1916, Rouen – 18 December 1994, Caen) was a French mathematician most remembered for Apéry's theorem, which states that is an irrational number.

See Riemann zeta function and Roger Apéry

Sieve of Eratosthenes

In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.

See Riemann zeta function and Sieve of Eratosthenes

Simon Stevin (journal)

Simon Stevin was a Dutch language academic journal in pure and applied mathematics, or Wiskunde as the field is known in Dutch.

See Riemann zeta function and Simon Stevin (journal)

Special values of L-functions

In mathematics, the study of special values of -functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely 1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \;. Riemann zeta function and special values of L-functions are zeta and L-functions.

See Riemann zeta function and Special values of L-functions

Statistics

Statistics (from German: Statistik, "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.

See Riemann zeta function and Statistics

Stieltjes constants

In mathematics, the Stieltjes constants are the numbers \gamma_k that occur in the Laurent series expansion of the Riemann zeta function: The constant \gamma_0. Riemann zeta function and Stieltjes constants are zeta and L-functions.

See Riemann zeta function and Stieltjes constants

String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

See Riemann zeta function and String theory

Symmetry

Symmetry in everyday life refers to a sense of harmonious and beautiful proportion and balance.

See Riemann zeta function and Symmetry

Tate's thesis

In number theory, Tate's thesis is the 1950 PhD thesis of completed under the supervision of Emil Artin at Princeton University. Riemann zeta function and Tate's thesis are zeta and L-functions.

See Riemann zeta function and Tate's thesis

Theta function

In mathematics, theta functions are special functions of several complex variables.

See Riemann zeta function and Theta function

Thue–Morse sequence

In mathematics, the Thue–Morse or Prouhet–Thue–Morse sequence is the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far.

See Riemann zeta function and Thue–Morse sequence

Triviality (mathematics)

In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces).

See Riemann zeta function and Triviality (mathematics)

Upper half-plane

In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with.

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Vinogradov's mean-value theorem

In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers.

See Riemann zeta function and Vinogradov's mean-value theorem

Weierstrass factorization theorem

In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes.

See Riemann zeta function and Weierstrass factorization theorem

Z function

In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half. Riemann zeta function and z function are zeta and L-functions.

See Riemann zeta function and Z function

Zero of a function

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) vanishes at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x).

See Riemann zeta function and Zero of a function

Zeros and poles

In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable.

See Riemann zeta function and Zeros and poles

Zeta

Zeta (uppercase Ζ, lowercase ζ; ζῆτα, label, classical or zē̂ta; zíta) is the sixth letter of the Greek alphabet.

See Riemann zeta function and Zeta

Zeta function regularization

In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. Riemann zeta function and zeta function regularization are zeta and L-functions.

See Riemann zeta function and Zeta function regularization

Zeta function universality

In mathematics, the universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well. Riemann zeta function and zeta function universality are zeta and L-functions.

See Riemann zeta function and Zeta function universality

ZetaGrid

ZetaGrid was at one time the largest distributed computing project, designed to explore the non-trivial roots of the Riemann zeta function, checking over one billion roots a day. Riemann zeta function and ZetaGrid are zeta and L-functions.

See Riemann zeta function and ZetaGrid

Zipf's law

Zipf's law is an empirical law that often holds, approximately, when a list of measured values is sorted in decreasing order.

See Riemann zeta function and Zipf's law

1 + 1 + 1 + 1 + ⋯

In mathematics,, also written,, or simply, is a divergent series.

See Riemann zeta function and 1 + 1 + 1 + 1 + ⋯

1 + 2 + 3 + 4 + ⋯

The infinite series whose terms are the natural numbers is a divergent series.

See Riemann zeta function and 1 + 2 + 3 + 4 + ⋯

3Blue1Brown

3Blue1Brown is a math YouTube channel created and run by Grant Sanderson.

See Riemann zeta function and 3Blue1Brown

References

[1] https://en.wikipedia.org/wiki/Riemann_zeta_function

Also known as Critical strip, Euler product formula, Euler zeta function, Euler-Riemann zeta function, Reimann Zeta function, Riemann functional equation, Riemann z-function, Riemann zeta, Riemann zeta function zeros, Riemann zeta-function, Riemann ζ function, Riemann ζ-function, Riemann's functional equation, Riemann's zeta function, Riemann-zeta function, Series of reciprocal powers, Trivial zero, Z(s), Ζ(s), Ζ(x).

, G. H. Hardy, Gamma function, Gauss–Kuzmin–Wirsing operator, Generalized Riemann hypothesis, Geometric series, Graduate Texts in Mathematics, Greek alphabet, Hankel contour, Harmonic series (mathematics), Harmonic series (music), Helmut Hasse, Holomorphic function, Hurwitz zeta function, Infinite product, Infinity, International Journal of Mathematics and Mathematical Sciences, Jacques Hadamard, John Edensor Littlewood, John Tate (mathematician), Jordan's totient function, Joseph Ser, Journal of Computational and Applied Mathematics, Journal of Number Theory, Kanakanahalli Ramachandra, Konrad Knopp, L-function, Lambert W function, Laurent series, Lehmer pair, Leonhard Euler, Lerch zeta function, Limit of a sequence, List of zeta functions, Mathematische Zeitschrift, Möbius function, Möbius inversion formula, Mean value theorem, Mellin transform, Meromorphic function, Modular form, Multiple zeta function, Multiplicative function, Musical tuning, Natural density, Odlyzko–Schönhage algorithm, On the Number of Primes Less Than a Given Magnitude, Oxford, Pafnuty Chebyshev, Peter Borwein, Physics, Poisson summation formula, Polygamma function, Polylogarithm, Power law, Prime number, Prime number theorem, Prime zeta function, Prime-counting function, Primorial, Probability theory, Proceedings of the American Mathematical Society, Proof of the Euler product formula for the Riemann zeta function, Pure mathematics, Quantum field theory, Ramanujan summation, Rational number, Rational zeta series, Real number, Regularization (physics), Renormalization, Residue (complex analysis), Riemann hypothesis, Riemann sphere, Riemann Xi function, Riemann–Siegel theta function, Roger Apéry, Sieve of Eratosthenes, Simon Stevin (journal), Special values of L-functions, Statistics, Stieltjes constants, String theory, Symmetry, Tate's thesis, Theta function, Thue–Morse sequence, Triviality (mathematics), Upper half-plane, Vinogradov's mean-value theorem, Weierstrass factorization theorem, Z function, Zero of a function, Zeros and poles, Zeta, Zeta function regularization, Zeta function universality, ZetaGrid, Zipf's law, 1 + 1 + 1 + 1 + ⋯, 1 + 2 + 3 + 4 + ⋯, 3Blue1Brown.

Riemann zeta function, the Glossary

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