Euler product formula for the Riemann zeta function

E54270

identity in analytic number theory mathematical formula

The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.

Aliases (2)

Euler product ×2
Euler products for Dirichlet L-functions ×1

Statements (48)

Predicate	Object
instanceOf	identity in analytic number theory → mathematical formula →
alternativeForm	log ζ(s) = ∑_{p} ∑_{k≥1} p^{−ks}/k for Re(s) > 1 →
assumes	s is a complex number with Re(s) > 1 →
category	Euler product →
consequence	knowledge of zeta on Re(s) > 1 encodes information about primes → primes determine the zeta function on Re(s) > 1 →
dependsOn	Euler’s method of rearranging absolutely convergent series → absolute convergence of the Dirichlet series for Re(s) > 1 →
domainOfValidity	complex numbers s with real part greater than 1 →
equivalentTo	sum over n ≥ 1 of 1/n^s equals product over primes p of 1/(1 − p^(−s)) for Re(s) > 1 →
expresses	Riemann zeta function as an infinite product over primes →
field	analytic number theory → number theory →
generalizedBy	Euler products for Dirichlet L-functions → Euler products for automorphic L-functions →
historicalDevelopment	discovered by Leonhard Euler in the 18th century →
implies	non-vanishing of the zeta function on the half-plane Re(s) > 1 → zeta function has no zeros for Re(s) > 1 →
importance	central in understanding the link between primes and analytic properties of zeta → fundamental tool in analytic number theory →
mathematicalObject	complex variable s → infinite product →
namedAfter	Leonhard Euler →
relatedConcept	Chebyshev functions → Hadamard product for entire functions → Mertens’ theorems → Riemann Hypothesis → prime number theorem →
relatesTo	Dirichlet L-functions → Dirichlet series → Riemann zeta function → distribution of prime numbers → fundamental theorem of arithmetic → logarithm of the zeta function → prime numbers → unique prime factorization →
shows	deep connection between primes and the zeta function → zeta function encodes information about primes →
standardForm	ζ(s) = ∏_{p prime} (1 − p^{−s})^{−1} for Re(s) > 1 →
usedFor	deriving estimates for the prime-counting function → establishing connections between primes and zeros of zeta → proofs related to the distribution of primes → proving results about multiplicative functions → studying analytic continuation of zeta and L-functions →
uses	infinite product over all prime numbers → multiplicativity of arithmetic functions → unique factorization of integers into primes →

Referenced by (4)

Subject (surface form when different)	Predicate
Euler product formula for the Riemann zeta function ("Euler product") →	category
Euler product formula for the Riemann zeta function ("Euler products for Dirichlet L-functions") →	generalizedBy
Leonhard Euler →	notableWork
Über die Anzahl der Primzahlen unter einer gegebenen Grösse ("Euler product") →	topic