Lecture

Primes in Arithmetic Progression

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture delves into the concept of primes in arithmetic progression, exploring the relationship between Dirichlet L-functions and characters. The instructor discusses the orthogonality of characters, the analytic continuation of L-functions, and the class number formula. By analyzing the behavior of L-functions at s=1, the lecture aims to demonstrate the divergence of the sum of 1 over p for p congruent to a modulo q. The proof involves showing that the L-function for non-trivial characters tends to infinity as s approaches 1. The lecture also touches upon Legendre symbols and their connection to real characters. Through a detailed analysis, the instructor illustrates the intricate interplay between characters, L-functions, and prime numbers.

Official source

Ontological neighbourhood

Mathematics

Number theory: Topics in number theory

Related concepts (32)

Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

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 functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers).

Ideal class group

In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK/PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K. The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group.

Dirichlet character

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function is a Dirichlet character of modulus (where is a positive integer) if for all integers and : that is, is completely multiplicative. (gcd is the greatest common divisor) that is, is periodic with period . The simplest possible character, called the principal character, usually denoted , (see Notation below) exists for all moduli: 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.

Formula for primes

In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be. A simple formula is for positive integer , where is the floor function, which rounds down to the nearest integer. By Wilson's theorem, is prime if and only if . Thus, when is prime, the first factor in the product becomes one, and the formula produces the prime number .

Related lectures (40)

Prime Numbers and Primality Testing

Covers prime numbers, RSA cryptography, and primality testing, including the Chinese Remainder Theorem and the Miller-Rabin test.

Number Theory: Division, Remainder, Congruence

Covers number theory, division, remainder, congruence, prime numbers, integer representation, and the Euclidean algorithm.

Multicorrelation Sequences and Primes

Explores multicorrelation sequences, primes, and their intricate connections in number theory and ergodic theory.

Rudiments of Number Theory

Introduces modulo arithmetic, Euclid's algorithm, and congruence in number theory.

Number Theory: More Facts about Primes