Mertens’ theorems

E300762

result in number theory theorem in analytic number theory

Mertens’ theorems are classical results in analytic number theory that give precise asymptotic estimates for sums involving the Möbius function and the reciprocals of primes, illuminating the distribution of primes and their connection to the Riemann zeta function.

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All labels observed (5)

Label	Occurrences
Mertens theorems	2
Mertens’ theorems canonical	2
Mertens’ first theorem	1
Mertens’ second theorem	1
Mertens’ third theorem	1

Statements (42)

Predicate	Object
instanceOf	result in number theory ⓘ theorem in analytic number theory ⓘ theorem in analytic number theory ⓘ theorem in analytic number theory ⓘ theorem in analytic number theory ⓘ
appliesTo	multiplicative arithmetic functions ⓘ prime numbers ⓘ
clarifies	connection between primes and the Riemann zeta function ⓘ
describes	asymptotic behavior of the product over primes (1 − 1/p) ⓘ asymptotic behavior of the sum of reciprocals of primes ⓘ
field	analytic number theory ⓘ number theory ⓘ
gives	precise asymptotic estimates for products over primes ⓘ precise asymptotic estimates for sums involving primes ⓘ
hasConsequence	estimates for partial Euler products ⓘ information about density of primes ⓘ refined bounds for sums over primes ⓘ
hasPart	Mertens’ theorems self-linksurface differs ⓘ surface form: Mertens’ first theorem Mertens’ theorems self-linksurface differs ⓘ surface form: Mertens’ second theorem Mertens’ theorems self-linksurface differs ⓘ surface form: Mertens’ third theorem
historicalPeriod	19th century mathematics ⓘ
involves	Euler–Mascheroni constant γ ⓘ surface form: Euler–Mascheroni constant Möbius function ⓘ iterated logarithm ⓘ natural logarithm ⓘ prime numbers ⓘ sums over primes ⓘ
mainTopic	Möbius function ⓘ Riemann zeta function ⓘ distribution of prime numbers ⓘ
namedAfter	Austrian mathematician Franz Mertens ⓘ Franz Mertens ⓘ
relatedTo	Chebyshev inequalities ⓘ surface form: Chebyshev’s theorems Dirichlet series ⓘ Mertens function ⓘ prime number theorem ⓘ surface form: Prime Number Theorem Riemann hypothesis ⓘ surface form: Riemann Hypothesis
statement	The product_{p \le x} (1 − 1/p) ~ e^{−γ}/log x as x → ∞ ⓘ The sum_{p \le x} (log p)/p = log x + O(1) as x → ∞ ⓘ The sum_{p \le x} 1/p = log log x + B + o(1) as x → ∞ for a constant B ⓘ
usesTool	complex analysis ⓘ properties of the Riemann zeta function ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Euler product formula for the Riemann zeta function → relatedConcept → Mertens’ theorems ⓘ

Multiplicative Number Theory → hasClassicResult → Mertens’ theorems ⓘ

this entity surface form: Mertens theorems

prime number theorem → relatedConcept → Mertens’ theorems ⓘ

Chebyshev functions → relatedTo → Mertens’ theorems ⓘ

this entity surface form: Mertens theorems

Mertens’ theorems → hasPart → Mertens’ theorems self-linksurface differs ⓘ

this entity surface form: Mertens’ first theorem

Mertens’ theorems → hasPart → Mertens’ theorems self-linksurface differs ⓘ

this entity surface form: Mertens’ second theorem

Mertens’ theorems → hasPart → Mertens’ theorems self-linksurface differs ⓘ

this entity surface form: Mertens’ third theorem