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Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prim ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Function
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by :\vartheta(x)=\sum_ \ln p where \ln denotes the natural logarithm, with the sum extending over all prime numbers that are less than or equal to . The second Chebyshev function is defined similarly, with the sum extending over all prime powers not exceeding : \psi(x) = \sum_\sum_\ln p=\sum_ \Lambda(n) = \sum_\left\lfloor\log_p x\right\rfloor\ln p, where is the von Mangoldt function. The Chebyshev functions, especially the second one , are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, (See the exact formula, below.) Both Chebyshev functions are asymptotic to , a statement equivalent to the prime number theorem. Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 seminal9-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin''. Overview This paper studies the prime-counting function using analytic methods. Although it is the only paper Riemann ever published on number theory, it contains ideas which influenced thousands of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory. Among the new definitions, ideas, and notation introduced: *The use of the Greek letter zeta (ζ) for a function previously mentioned by Euler *The analytic continuation of this zeta functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand's Postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3, there always exists at least one prime number p with :n < p < 2n - 2. A less restrictive formulation is: for every , there is always at least one prime such that : Another formulation, where is the -th prime, is: for : This statement was first d in 1845 by (1822–1900). Bertrand himself verified his statement for all integers . His conjecture was completely [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage, and Chebyshev bias. Transcription The surname Chebyshev has been transliterated in several different ways, like Tchebichef, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, Tschebyscheff, Čebyčev, Čebyšev, Chebysheff, Chebychov, Chebyshov (according to native Russian speakers, this one provides the closest pronunciation in English to the correct pronunciation in old Russian) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Integral
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value x. Integral representation The logarithmic integral has an integral representation defined for all positive real numbers ≠ 1 by the definite integral : \operatorname(x) = \int_0^x \frac. Here, denotes the natural logarithm. The function has a singularity at , and the integral for is interpreted as a Cauchy principal value, : \operatorname(x) = \lim_ \left( \int_0^ \frac + \int_^x \frac \right). Offset logarithmic integral The offset logarithmic integral or Eulerian logarithmic integral is defined as : \operatorname(x) = \int_2^x \frac = \operatorname(x) - \operatorname(2). As s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function. Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in particular for results named after him. Biography Early life (1805–1822) Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of Vienna in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from Richelette (or more likely Richelle), a small community north e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Constant
Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function \pi(x). Its value is now known to be 1. Examination of available numerical evidence for known primes led Legendre to suspect that \pi(x) satisfies an approximate formula. Legendre conjectured in 1808 that : \pi(x) = \frac where \lim_ B(x) = 1.08366 .... Or similarly, :\lim_ \left( \ln(n) - \right)= B where ''B'' is Legendre's constant. He guessed ''B'' to be about 1.08366, but regardless of its exact value, the existence of ''B'' implies the prime number theorem. Pafnuty Chebyshev proved in 1849 that if the limit ''B'' exists, it must be equal to 1. An easier proof was given by Pintz in 1980. It is an immediate consequence of the prime number theorem, under the precise form with an explicit estimate of the error term : \pi(x)= (x) + O \left(x e^\right) \quad\text x \to \infty (for some positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. Life Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media. This treatise also brought him to the attention of Lagrange. The ''Académie des sciences'' made Legendre an adjoint member in 1783 and an associate in 1785. In 1789, he was elected a Fellow of the Royal Society. He assisted with the Anglo-French Sur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
