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Siegel–Walfisz Theorem
In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet's theorem on primes in arithmetic progressions. Statement Define :\psi(x;q,a) = \sum_\Lambda(n), where \Lambda denotes the von Mangoldt function, and let ''φ'' denote Euler's totient function. Then the theorem states that given any real number ''N'' there exists a positive constant ''C''''N'' depending only on ''N'' such that :\psi(x;q,a)=\frac+O\left(x\exp\left(-C_N(\log x)^\frac\right)\right), whenever (''a'', ''q'') = 1 and :q\le(\log x)^N. Remarks The constant ''C''''N'' is not effectively computable because Siegel's theorem is ineffective. From the theorem we can deduce the following bound regarding the prime number theorem for arithmetic progressions: If, for (''a'', ''q'') = 1, by \pi(x;q,a) we denote the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. *Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Mangoldt Function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mangoldt function, denoted by , is defined as :\Lambda(n) = \begin \log p & \textn=p^k \text p \text k \ge 1, \\ 0 & \text \end The values of for the first nine positive integers (i.e. natural numbers) are :0 , \log 2 , \log 3 , \log 2 , \log 5 , 0 , \log 7 , \log 2 , \log 3, which is related to . Properties The von Mangoldt function satisfies the identityApostol (1976) p.32Tenenbaum (1995) p.30 :\log(n) = \sum_ \Lambda(d). The sum is taken over all integers that divide . This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to . For example, consider the case . Then :\begin \sum_ \Lambda(d) &= \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(4) + \Lambda(6) + \Lambda(12) \\ &= \ ... [...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 su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem For Arithmetic Progressions
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 prime (), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Effective Results In Number Theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable. Where it is asserted that some list of integers is finite, the question is whether in principle the list could be printed out after a machine computation. Littlewood's result An early example of an ineffective result was J. E. Littlewood's theorem of 1914, that in the prime number theorem the differences of both ψ(''x'') and π(''x'') with their asymptotic estimates change sign infinitely often. In 1933 Stanley Skewes obtained an effective upper bound for the first sign change, now known as Skewes' number. In more detail, writing for a numerical sequence ''f'' (''n''), an ''effective'' result about its changing sign infinitely often would be a theorem including, for every value of ''N'', a value ''M'' > ''N'' such that ''f'' ('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RSA e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet's Theorem On Primes In Arithmetic Progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is also a positive integer. In other words, there are infinitely many primes that are congruent to ''a'' modulo ''d''. The numbers of the form ''a'' + ''nd'' form an arithmetic progression :a,\ a+d,\ a+2d,\ a+3d,\ \dots,\ and Dirichlet's theorem states that this sequence contains infinitely many prime numbers. The theorem, named after Peter Gustav Lejeune Dirichlet, extends Euclid's theorem that there are infinitely many prime numbers. Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same proportions of primes. Equivalently, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Walfisz
Arnold Walfisz (2 July 1892 – 29 May 1962) was a Jewish-Polish mathematician working in analytic number theory. Life After the ''Abitur'' in Warsaw (Poland), Arnold Walfisz studied (1909−14 and 1918−21) in Germany at Munich, Berlin, Heidelberg and Göttingen. Edmund Landau was his doctoral-thesis supervisor at the University of Göttingen. Walfisz lived in Wiesbaden from 1922 through 1927, then he returned to Warsaw, worked at an insurance company and at the mathematical institute of the university (habilitation in 1930). In 1935, together with , he founded the mathematical journal ''Acta Arithmetica''. In 1936, Walfisz became professor at the University of Tbilisi in the nation of Georgia (at the time a part of the Soviet Union). He wrote approximately 100 mathematical articles and three books. Work By using a theorem by Carl Ludwig Siegel providing an upper bound for the real zeros (see Siegel zero) of Dirichlet L-functions formed with real non-principal characters, W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primes In Arithmetic Progression
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a_n = 3 + 4n for 0 \le n \le 2. According to the Green–Tao theorem, there exist arbitrarily long sequences of primes in arithmetic progression. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form an + b, where ''a'' and ''b'' are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For integer ''k'' ≥ 3, an AP-''k'' (also called PAP-''k'') is any sequence of ''k'' primes in arithmetic progression. An AP-''k'' can be written as ''k'' primes of the form ''a''·''n'' + ''b'', for fixed integers ''a'' (called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
