Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2. The Dedekind zeta function is named for Richard Dedekind who introduced it in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie. Let K be an algebraic number field. Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series where I ranges through the non-zero ideals of the ring of integers OK of K and NK/Q(I) denotes the absolute norm of I (which is equal to both the index [OK : I] of I in OK or equivalently the cardinality of quotient ring OK / I). This sum converges absolutely for all complex numbers s with real part Re(s) > 1. In the case K = Q, this definition reduces to that of the Riemann zeta function. The Dedekind zeta function of has an Euler product which is a product over all the non-zero prime ideals of This is the expression in analytic terms of the uniqueness of prime factorization of ideals in . For is non-zero. Erich Hecke first proved that ζK(s) has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at s = 1. The residue at that pole is given by the analytic class number formula and is made up of important arithmetic data involving invariants of the unit group and class group of K. The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let ΔK denote the discriminant of K, let r1 (resp. r2) denote the number of real places (resp.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (1)

Related concepts (24)

Related courses (6)

Related lectures (44)

MATH-482: Algebraic number theory

Algebraic number theory is the study of the properties of solutions of polynomial equations with integral coefficients; Starting with concrete problems, we then introduce more general notions like alg

MATH-317: Galois theory

Ce cours commence par un rappel de la théorie de Galois vue en 2ème année : les extensions de corps et la correspondance de Galois. Ensuite, on présente les applications et approfondissements.

MATH-313: Introduction to analytic number theory

The aim of this course is to present the basic techniques of analytic number theory.

The fourth moment of individual Dirichlet L-functions on the critical line

Berke Topacogullari

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the

2020

In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified. The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K.

Galois Theory: The Galois Correspondence MATH-317: Galois theory

Explores the Galois correspondence and solvability by radicals in polynomial equations.

Meromorphic Functions & Differentials MATH-410: Riemann surfaces

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Galois Theory: Solvability and Radical Extensions MATH-317: Galois theory

Explores solvability by radicals in Galois theory and the Galois/Abel criterion for solvability.