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# Weyl's Inequality in Number Theory

Weyl's inequality in number theory

In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies | c a / q | t q 2 , displaystyle |c-a/q|leq tq^-2, for some t greater than or equal to 1, then for any positive real number displaystyle scriptstyle varepsilon one has x = M M N exp ( 2 i f ( x ) ) = O ( N 1 ( t q 1 N t N k 1 q N k ) 2 1 k ) as N . displaystyle sum _x=M^MNexp(2pi if(x))=Oleft(N^1varepsilon left(t over q1 over Nt over N^k-1q over N^k

ight)^2^1-k

ight)text as Nto infty . This inequality will only be useful when q

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Journal of Number Theory

The Journal of Number Theory is a bimonthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. Woods, and H. Zassenhaus (Ohio State University). It is currently published monthly by Elsevier and the editor-in-chief is Dorian Goldfeld (Columbia University). According to the Journal Citation Reports, the journal has a 2017 impact factor of 0.774.

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Contributions to number theory

Frobenius introduced a canonical way of turning primes into conjugacy classes in Galois groups over Q. Specifically, if K/Q is a finite Galois extension then to each (positive) prime p which does not ramify in K and to each prime ideal P lying over p in K there is a unique element g of Gal(K/Q) satisfying the condition g(x) = xp (mod P) for all integers x of K. Varying P over p changes g into a conjugate (and every conjugate of g occurs in this way), so the conjugacy class of g in the Galois group is canonically associated to p. This is called the Frobenius conjugacy class of p and any element of the conjugacy class is called a Frobenius element of p. If we take for K the mth cyclotomic field, whose Galois group over Q is the units modulo m (and thus is abelian, so conjugacy classes become elements), then for p not dividing m the Frobenius class in the Galois group is p mod m. From this point of view, the distribution of Frobenius conjugacy classes in Galois groups over Q (or, more generally, Galois groups over any number field) generalizes Dirichlet's classical result about primes in arithmetic progressions. The study of Galois groups of infinite-degree extensions of Q depends crucially on this construction of Frobenius elements, which provides in a sense a dense subset of elements which are accessible to detailed study.

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List of algebraic number theory topics

This is a list of algebraic number theory topics

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Number theory

BrhmasphuasiddhntaBrahmagupta (628)Brahmagupta's Brhmasphuasiddhnta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. The current system of the four fundamental operations (addition, subtraction, multiplication and division) based on the Hindu-Arabic number system also first appeared in Brahmasphutasiddhanta. It was also one of the first texts to provide concrete ideas on positive and negative numbers. De fractionibus continuis dissertatioLeonhard Euler (1744)First presented in 1737, this paper provided the first then-comprehensive account of the properties of continued fractions. It also contains the first proof that the number e is irrational. Recherches d'ArithmtiqueJoseph Louis Lagrange (1775)Developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form a x 2 b y 2 c x y displaystyle ax^2by^2cxy . This included a reduction theory for binary quadratic forms, where he proved that every form is equivalent to a certain canonically chosen reduced form. Disquisitiones ArithmeticaeCarl Friedrich Gauss (1801)The Disquisitiones Arithmeticae is a profound and masterful book on number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds many important new results of his own. Among his contributions was the first complete proof known of the Fundamental theorem of arithmetic, the first two published proofs of the law of quadratic reciprocity, a deep investigation of binary quadratic forms going beyond Lagrange's work in Recherches d'Arithmtique, a first appearance of Gauss sums, cyclotomy, and the theory of constructible polygons with a particular application to the constructibility of the regular 17-gon. Of note, in section V, article 303 of Disquisitiones, Gauss summarized his calculations of class numbers of imaginary quadratic number fields, and in fact found all imaginary quadratic number fields of class numbers 1, 2, and 3 (confirmed in 1986) as he had conjectured. In section VII, article 358, Gauss proved what can be interpreted as the first non-trivial case of the Riemann Hypothesis for curves over finite fields (the Hasse-Weil theorem). "Beweis des Satzes, da jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthlt"Peter Gustav Lejeune Dirichlet (1837)Pioneering paper in analytic number theory, which introduced Dirichlet characters and their L-functions to establish Dirichlet's theorem on arithmetic progressions. In subsequent publications, Dirichlet used these tools to determine, among other things, the class number for quadratic forms. "ber die Anzahl der Primzahlen unter einer gegebenen Grsse"Bernhard Riemann (1859)"ber die Anzahl der Primzahlen unter einer gegebenen Grsse" (or "On the Number of Primes Less Than a Given Magnitude") is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. Although it is the only paper he ever published on number theory, it contains ideas which influenced dozens 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. It also contains the famous Riemann Hypothesis, one of the most important open problems in mathematics. Vorlesungen ber ZahlentheoriePeter Gustav Lejeune Dirichlet and Richard DedekindVorlesungen ber Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P. G. Lejeune Dirichlet and R. Dedekind, and published in 1863. The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert. Dirichlet does not explicitly recognise the concept of the group that is central to modern algebra, but many of his proofs show an implicit understanding of group theory ZahlberichtDavid Hilbert (1897)Unified and made accessible many of the developments in algebraic number theory made during the nineteenth century. Although criticized by Andr Weil (who stated "more than half of his famous Zahlbericht is little more than an account of Kummer's number-theoretical work, with inessential improvements") and Emmy Noether, it was highly influential for many years following its publication. Fourier Analysis in Number Fields and Hecke's Zeta-FunctionsJohn Tate (1950)Generally referred to simply as Tate's Thesis, Tate's Princeton PhD thesis, under Emil Artin, is a reworking of Erich Hecke's theory of zeta- and L-functions in terms of Fourier analysis on the adeles. The introduction of these methods into number theory made it possible to formulate extensions of Hecke's results to more general L-functions such as those arising from automorphic forms. "Automorphic Forms on GL(2)"Herv Jacquet and Robert Langlands (1970)This publication offers evidence towards Langlands' conjectures by reworking and expanding the classical theory of modular forms and their L-functions through the introduction of representation theory. "La conjecture de Weil. I."Pierre Deligne (1974)Proved the Riemann hypothesis for varieties over finite fields, settling the last of the open Weil conjectures. "Endlichkeitsstze fr abelsche Varietten ber Zahlkrpern"Gerd Faltings (1983)Faltings proves a collection of important results in this paper, the most famous of which is the first proof of the Mordell conjecture (a conjecture dating back to 1922). Other theorems proved in this paper include an instance of the Tate conjecture (relating the homomorphisms between two abelian varieties over a number field to the homomorphisms between their Tate modules) and some finiteness results concerning abelian varieties over number fields with certain properties. "Modular Elliptic Curves and Fermat's Last Theorem"Andrew Wiles (1995)This article proceeds to prove a special case of the Shimura-Taniyama conjecture through the study of the deformation theory of Galois representations. This in turn implies the famed Fermat's Last Theorem. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory. The geometry and cohomology of some simple Shimura varietiesMichael Harris and Richard Taylor (2001)Harris and Taylor provide the first proof of the local Langlands conjecture for GL(n). As part of the proof, this monograph also makes an in depth study of the geometry and cohomology of certain Shimura varieties at primes of bad reduction. "Le lemme fondamental pour les algbres de Lie"Ng Bo ChuNg Bo Chu proved a long-standing unsolved problem in the classical Langlands program, using methods from the Geometric Langlands program.

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