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Burgess Bound
In analytic number theory, the Burgess inequality (also called the Burgess bound) is an inequality that provides an upper bound for character sums :S_(N,H):=\sum\limits_ \chi(n) where \chi is a Dirichlet character modulo a cube free p\in\mathbb that is not the principal character \chi_0. The inequality was proven in 1963 along with a series of related inequalities, by the British mathematician David Allan Burgess. It provides a better estimate for small character sums than the Pólya–Vinogradov inequality from 1918. More recent results have led to refinements and generalizations of the Burgess bound. Burgess inequality A number is called ''cube free'' if it is not divisible by any cubic number x^3 except \pm 1. Define r\in \mathbb with r\geq 2 and \varepsilon>0. Let \chi be a Dirichlet character modulo p\in\mathbb that is not a principal character. For two N,H\in\mathbb, define the character sum :S_(N,H):=\sum\limits_ \chi(n). If either p is cube free or r\leq 3 ... [...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 numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by and , respectively the less-than sign, less-than and greater-than sign, greater-than signs). Notation There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' or ''a'' ≦ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . and other numbers ''x'' such that would be an upper bound for ''S''. The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character Sum
In mathematics, a character sum is a sum \sum \chi(n) of values of a Dirichlet character χ ''modulo'' ''N'', taken over a given range of values of ''n''. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue ''modulo'' ''N''. Character sums are often closely linked to exponential sums by the Gauss sums (this is like a finite Mellin transform). Assume χ is a non-principal Dirichlet character to the modulus ''N''. Sums over ranges The sum taken over all residue classes mod ''N'' is then zero. This means that the cases of interest will be sums \Sigma over relatively short ranges, of length ''R'' < ''N'' say, : A fundamental improvement on the trivial estimate is the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi: \mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: # \chi(ab) = \chi(a)\chi(b); that is, \chi is completely multiplicative. # \chi(a) \begin =0 &\text \gcd(a,m)>1\\ \ne 0&\text\gcd(a,m)=1. \end (gcd is the greatest common divisor) # \chi(a + m) = \chi(a); that is, \chi is periodic with period m. The simplest possible character, called the principal character, usually denoted \chi_0, (see Notation below) exists for all moduli: : \chi_0(a)= \begin 0 &\text \gcd(a,m)>1\\ 1 &\text \gcd(a,m)=1. \end The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions. Notation \phi(n) is Euler's totient function. \zeta_n is a complex primitive n-th root of unity: : \zeta_n^n=1, but \zeta_n\ne 1, \ze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Northwestern Europe, off the coast of European mainland, the continental mainland. It comprises England, Scotland, Wales and Northern Ireland. The UK includes the island of Great Britain, the north-eastern part of the island of Ireland, and most of List of islands of the United Kingdom, the smaller islands within the British Isles, covering . Northern Ireland shares Republic of Ireland–United Kingdom border, a land border with the Republic of Ireland; otherwise, the UK is surrounded by the Atlantic Ocean, the North Sea, the English Channel, the Celtic Sea and the Irish Sea. It maintains sovereignty over the British Overseas Territories, which are located across various oceans and seas globally. The UK had an estimated population of over 68.2 million people in 2023. The capital and largest city of both England and the UK is London. The cities o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Allan Burgess
David (; , "beloved one") was a king of ancient Israel and Judah and the third king of the United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Aramaic-inscribed stone erected by a king of Aram-Damascus in the late 9th/early 8th centuries BCE to commemorate a victory over two enemy kings, contains the phrase (), which is translated as "House of David" by most scholars. The Mesha Stele, erected by King Mesha of Moab in the 9th century BCE, may also refer to the "House of David", although this is disputed. According to Jewish works such as the ''Seder Olam Rabbah'', ''Seder Olam Zutta'', and ''Sefer ha-Qabbalah'' (all written over a thousand years later), David ascended the throne as the king of Judah in 885 BCE. Apart from this, all that is known of David comes from biblical literature, the historicity of which has been extensively challenged,Writing and Rewriting the Story of Solomon in Ancient Israel; by Isaac Kalimi; page 32; Cambr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pólya–Vinogradov Inequality
In mathematics, a character sum is a sum \sum \chi(n) of values of a Dirichlet character χ ''modulo'' ''N'', taken over a given range of values of ''n''. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue ''modulo'' ''N''. Character sums are often closely linked to exponential sums by the Gauss sums (this is like a finite Mellin transform). Assume χ is a non-principal Dirichlet character to the modulus ''N''. Sums over ranges The sum taken over all residue classes mod ''N'' is then zero. This means that the cases of interest will be sums \Sigma over relatively short ranges, of length ''R'' < ''N'' say, : A fundamental improvement on the trivial estimate is the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henryk Iwaniec
Henryk Iwaniec (born October 9, 1947) is a Polish-American mathematician, and since 1987 a professor at Rutgers University. He is a member of the American Academy of Arts and Sciences and Polish Academy of Sciences. He has made important contributions to analytic and algebraic number theory as well as harmonic analysis. He is the recipient of Cole Prize (2002), Steele Prize (2011), and Shaw Prize (2015). Background and education Iwaniec studied at the University of Warsaw, where he got his PhD in 1972 under Andrzej Schinzel. He then held positions at the Institute of Mathematics of the Polish Academy of Sciences until 1983 when he left Poland. He held visiting positions at the Institute for Advanced Study, University of Michigan, and University of Colorado Boulder before being appointed Professor of Mathematics at Rutgers University. He is a citizen of both Poland and the United States. He and mathematician Tadeusz Iwaniec are twin brothers. Work Iwaniec studies both s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emmanuel Kowalski
Immanuel or Emmanuel (, "God swith us"; Koine Greek: ) is a Hebrew name that appears in the Book of Isaiah (7:14) as a sign that God will protect the House of David. The Gospel of Matthew ( Matthew 1:22 –23) interprets this as a prophecy of the birth of the Messiah and the fulfillment of Scripture in the person of Jesus. ''Immanuel'' "God ( El) with us" is one of the "symbolic names" used by Isaiah, alongside Shearjashub, Maher-shalal-hash-baz, or Pele-joez-el-gibbor-abi-ad-sar-shalom. It has no particular meaning in Jewish messianism. In Christian theology by contrast, based on its use in Isaiah 7:14, the name has come to be read as a prophecy of the Christ, following Matthew 1:23, where ''Immanuel'' () is translated as (KJV: "God with us"), and also Luke 7:14–16 after the raising of the dead man in Nain, where it was rumoured throughout all Judaea that "God has visited his people" (KJV). Isaiah 7–8 Summary The setting is the Syro-Ephraimite War, 735-734 BCE, w ... [...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 numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
