|
Elliott–Halberstam Conjecture
In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated the conjecture in 1968. Stating the conjecture requires some notation. Let \pi(x), the prime-counting function, denote the number of primes less than or equal to x. If q is a positive integer and a is coprime to q, we let \pi(x;q,a) denote the number of primes less than or equal to x which are equal to a modulo q. Dirichlet's theorem on primes in arithmetic progressions then tells us that : \pi(x;q,a) \approx \frac where \varphi is Euler's totient function. If we then define the error function : E(x;q) = \max_ \left, \pi(x;q,a) - \frac\ where the max is taken over all a coprime to q, then the Elliott–Halberstam conjecture is the assertion that for every \theta 0 there exists a constant C > 0 such that : \sum_ E(x;q) \le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Riemann Hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these ''L''-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case). Global ''L''-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothes ... [...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] |
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] |
Sexy Prime
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and . The term "sexy prime" is a pun stemming from the Latin word for six: . If or (where is the lower prime) is also prime, then the sexy prime is part of a prime triplet. In August 2014 the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin or sexy primes. Primorial ''n''# notation As used in this article, # stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ . Types of groupings Sexy prime pairs The sexy primes (sequences and in OEIS) below 500 are: :(5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barban–Davenport–Halberstam Theorem
In mathematics, the Barban–Davenport–Halberstam theorem is a statement about the distribution of prime numbers in an arithmetic progression. It is known that in the long run primes are distributed equally across possible progressions with the same difference. Theorems of the Barban–Davenport–Halberstam type give estimates for the error term, determining how close to uniform the distributions are. Statement Let ''a'' be coprime to ''q'' and :\vartheta(x;q,a) = \sum_ \log p \ be a weighted count of primes in the arithmetic progression ''a'' mod ''q''. We have :\vartheta(x;q,a) = \frac + E(x;q,a) \ where ''φ'' is Euler's totient function and the error term ''E'' is small compared to ''x''. We take a sum of squares of error terms :V(x,Q) = \sum_ \sum_ , E(x;q,a), ^2 \ . Then we have :V(x,Q) = O(Q x \log x) + O(x^2 (\log x)^) \ for 1 \leq Q \leq x and every positive ''A'', where ''O'' is Landau's Big O notation. This form of the theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Elliott–Halberstam Conjecture
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Maynard (mathematician)
James Alexander Maynard (born 10 June 1987) is an English mathematician working in analytic number theory and in particular the theory of prime numbers. In 2017, he was appointed Research Professor at Oxford. Maynard is a fellow of St John's College, Oxford. He was awarded the Fields Medal in 2022. Biography Maynard attended King Edward VI Grammar School, Chelmsford in Chelmsford, England. After completing his bachelor's and master's degrees at Queens' College, University of Cambridge in 2009, Maynard obtained his D.Phil. from University of Oxford at Balliol College in 2013 under the supervision of Roger Heath-Brown. He then became a Fellow by Examination at Magdalen College, Oxford. For the 2013–2014 year, Maynard was a CRM-ISM postdoctoral researcher at the University of Montreal. In November 2013, Maynard gave a different proof of Yitang Zhang's theorem that there are bounded gaps between primes, and resolved a longstanding conjecture by showing that for any m there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cem Yıldırım
Cem Yalçın Yıldırım (born 8 July 1961) is a Turkish mathematician who specializes in number theory. He obtained his B.Sc from Middle East Technical University in Ankara, Turkey and his PhD from the University of Toronto in 1990. His advisor was John Friedlander. He is currently a faculty member at Boğaziçi University in Istanbul, Turkey. In 2005(), with Dan Goldston and János Pintz, he proved, that for any positive number ''ε'' there exist primes ''p'' and ''p''′ such that the difference between ''p'' and ''p''′ is smaller than ''ε'' log ''p''. Formally; :\liminf_\frac=0 where ''p''''n'' denotes the ''n''th prime number. In other words, for every ''c'' > 0, there exist infinitely many pairs of consecutive primes ''p''''n'' and ''p''''n''+1 which are closer to each other than the average distance between consecutive primes by a factor of ''c'', i.e., ''p''''n''+1 − ''p''''n'' < ''c'' log ''p''''n''. This r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
János Pintz
János Pintz (born 20 December 1950 in Budapest) is a Hungary, Hungarian mathematician working in analytic number theory. He is a fellow of the Alfréd Rényi Institute of Mathematics, Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences. In 2014, he received the Cole Prize. Mathematical results Pintz is best known for proving in 2005 (with Daniel Goldston and Cem Yıldırım) that :: \liminf_\frac=0 where p_n denotes the ''n''th prime number. In other words, for every ε > 0, there exist infinitely many pairs of consecutive primes ''p''''n'' and ''p''''n''+1 that are closer to each other than the average distance between consecutive primes by a factor of ε, i.e., ''p''''n''+1 − ''p''''n'' < ε log ''p''''n''. This result was originally reported in 2003 by Daniel Goldston and Cem Yıldırım but was later retracted. Pintz joined the team and comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
