## Prime numbers and L-functions           Video in TIB AV-Portal: Prime numbers and L-functions

 Title Prime numbers and L-functions Title of Series International Congress of Mathematicians, Madrid 2006 Number of Parts 33 Author License CC Attribution 3.0 Germany:You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. Identifiers 10.5446/15955 (DOI) Publisher Release Date 2006 Language English

 Subject Area Mathematics Abstract The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk I will show a panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory. Keywords prime numbers L-functions Point (geometry) Greatest element Letterpress printing Function (mathematics) Student's t-test Mereology Prime number Theory Hypothesis Explosion Different (Kate Ryan album) Number theory Modulform Analytic number theory Theorem Körper <Algebra> Cumulative distribution function Prime number theorem Perturbation theory Cartesian coordinate system Numerical analysis Degree (graph theory) Arithmetic progression Fundamental theorem of algebra Sinc function
Statistical hypothesis testing Cumulative distribution function Mathematics Process (computing) Film editing Length Algebraic structure Number theory Nichtlineares Gleichungssystem Function (mathematics) Game theory Prime number
Axiom of choice Logical constant Statistical hypothesis testing Complex (psychology) Group action Basis (linear algebra) INTEGRAL Multiplication sign Group theory Parameter (computer programming) Function (mathematics) Prime number Mereology Grothendieck topology Integer factorization Group representation Mathematics Estimator Sign (mathematics) Square number Number theory Extension (kinesiology) Social class Predictability Cumulative distribution function Covering space Area Primality test Moment (mathematics) Sampling (statistics) Complete metric space Arithmetic mean Vector space Right angle Cycle (graph theory) Arithmetic progression Resultant Metre Trail Functional (mathematics) Existence Observational study Adaptive behavior Graph coloring Product (business) Hypothesis Element (mathematics) Population density Exponential function Term (mathematics) Well-formed formula Average Operator (mathematics) Modulform Module (mathematics) Grundlagen der Mathematik Projective plane Model theory Prime number theorem Algebraic structure Total S.A. Line (geometry) Evolute Rectangle Limit (category theory) Cartesian coordinate system Film editing Combinatory logic Principle of relativity Theory of everything Object (grammar)
Group action INTEGRAL State of matter Multiplication sign Range (statistics) Group theory Function (mathematics) Mereology Prime number Different (Kate Ryan album) Square number Number theory Series (mathematics) Social class Cumulative distribution function Predictability Block (periodic table) Sequence Proof theory Arithmetic mean Uniformer Raum Arithmetic progression Point (geometry) Metre Trail Functional (mathematics) Transformation (genetics) Event horizon Hypothesis Inclusion map Natural number Well-formed formula Term (mathematics) Profil (magazine) Operator (mathematics) Queue (abstract data type) Modulform Devolution (biology) Associative property Set theory Wave function Model theory Mathematical analysis Algebraic structure Multilateration Incidence algebra Limit (category theory) Cartesian coordinate system Voting Logic Network topology Object (grammar)
Principal ideal Statistical hypothesis testing Randomization Group action Building Greatest element State of matter Multiplication sign Decision theory Group theory Function (mathematics) Parameter (computer programming) Prime number Mereology Explosion Estimator Mathematics Sign (mathematics) Many-sorted logic Square number Number theory Series (mathematics) Social class Cumulative distribution function Area Closed set Sampling (statistics) Physicalism Staff (military) Lattice (order) Sequence Measurement Arithmetic mean Resultant Directed graph Point (geometry) Computer programming Trail Functional (mathematics) Inverse function Existence Perfect group Parity (mathematics) Barrelled space Event horizon Product (business) Power (physics) Population density Exponential function Well-formed formula Natural number Operator (mathematics) Modulform Integer Genetic programming Nichtlineares Gleichungssystem Set theory Addition Standard deviation Dependent and independent variables Projective plane Prime number theorem Mathematical analysis Volume (thermodynamics) Line (geometry) Limit (category theory) Cartesian coordinate system Celestial sphere Combinatory logic Principle of relativity Theory of everything Game theory Object (grammar)
Point (geometry) Statistical hypothesis testing Complex (psychology) Functional (mathematics) Group action Existence State of matter Multiplication sign Group theory Function (mathematics) Mereology Abundant number Integer factorization Power (physics) Subset Explosion Frequency Estimator Well-formed formula Natural number Different (Kate Ryan album) Number theory Modulform Energy level Genetic programming Körper <Algebra> Series (mathematics) Associative property Set theory Position operator Condition number Social class Cumulative distribution function Multiplication Moment (mathematics) Lattice (order) Measurement Degree (graph theory) Category of being Arithmetic mean Voting Cube Order (biology) Principle of relativity Theory of everything Arithmetic progression
Standard deviation Cross-correlation
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