Euler–Mascheroni constant γ

E596525

mathematical constant

The Euler–Mascheroni constant γ is a mathematical constant that arises in analysis and number theory, defined as the limiting difference between the harmonic series and the natural logarithm.

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All labels observed (2)

Label	Occurrences
Euler–Mascheroni constant	1
Euler–Mascheroni constant γ canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T6482593 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Euler–Mascheroni constant γ
Context triple: [Gamma function, containsConstant, Euler–Mascheroni constant γ]

A. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
B. Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
C. Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
D. Khinchin–Lévy constant
The Khinchin–Lévy constant is a mathematical constant arising in metric number theory and continued fractions, describing the typical exponential growth rate of the denominators of convergents for almost all real numbers.
E. Pi
Pi is a 1998 psychological thriller film directed by Darren Aronofsky that explores obsession, mathematics, and mysticism through the story of a reclusive number theorist.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Euler–Mascheroni constant γ
Target entity description: The Euler–Mascheroni constant γ is a mathematical constant that arises in analysis and number theory, defined as the limiting difference between the harmonic series and the natural logarithm.

A. Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
B. Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
C. Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
D. Khinchin–Lévy constant
The Khinchin–Lévy constant is a mathematical constant arising in metric number theory and continued fractions, describing the typical exponential growth rate of the denominators of convergents for almost all real numbers.
E. Pi
Pi is a 1998 psychological thriller film directed by Darren Aronofsky that explores obsession, mathematics, and mysticism through the story of a reclusive number theorist.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	mathematical constant ⓘ
alternativeName	Euler constant NERFINISHED ⓘ Mascheroni constant NERFINISHED ⓘ
appearsIn	analysis ⓘ analytic number theory ⓘ asymptotic analysis of harmonic numbers ⓘ number theory ⓘ special functions ⓘ
approximateDecimalExpansion	0.57721566490153286060651209008240243104215933593992… ⓘ
approximateValue	0.57721 ⓘ 0.5772156649 ⓘ
conjecturedProperty	believed to be irrational ⓘ believed to be transcendental ⓘ
definition	lim_{n→∞}(1 + 1/2 + 1/3 + … + 1/n − ln n) ⓘ
discoveredBy	Leonhard Euler NERFINISHED ⓘ
domain	analytic number theory ⓘ real analysis ⓘ
historicalNote	further investigated by Lorenzo Mascheroni ⓘ studied by Euler in the 18th century ⓘ
integralRepresentation	γ = ∫_0^1 (1 − H_x) dx where H_x is analytically continued ⓘ γ = ∫_0^∞ (e^{−x}/x − e^{−x}/(1 − e^{−x})) dx ⓘ
namedAfter	Leonhard Euler NERFINISHED ⓘ Lorenzo Mascheroni NERFINISHED ⓘ
openProblem	It is unknown whether γ is algebraic or transcendental ⓘ It is unknown whether γ is rational or irrational ⓘ
relation	H_n = ln n + γ + 1/(2n) + O(1/n^2) ⓘ H_n = ln n + γ + o(1) as n→∞ ⓘ appears in Mertens' theorems ⓘ appears in estimates for the average order of arithmetic functions ⓘ appears in the Laurent expansion of ζ(s) at s = 1 ⓘ appears in the asymptotic expansion of the factorial via Γ function ⓘ appears in the constant term of many asymptotic expansions involving logarithms ⓘ ln n! = n ln n − n + (1/2) ln(2πn) + γ/(12n) + O(1/n^3) ⓘ Γ'(1) = −γ ⓘ γ = lim_{n→∞}(H_n − ln n) ⓘ γ = lim_{n→∞}(∑_{k=1}^n 1/k − ln n) ⓘ γ = −Γ'(1) ⓘ γ = −ψ(1) ⓘ γ = ∫_0^1 (1 − 1/ln x) dx is incorrect (divergent integral) ⓘ γ = ∫_1^∞ (1/⌊x⌋ − 1/x) dx ⓘ ζ(s) = 1/(s − 1) + γ + O(s − 1) as s→1 ⓘ ψ(x) = d/dx ln Γ(x) and ψ(1) = −γ ⓘ ∏_{p≤x} (1 − 1/p) ∼ e^{−γ}/ln x as x→∞ ⓘ ∑_{p≤x} 1/p = ln ln x + B_1 + o(1) where B_1 is related to γ ⓘ
seriesRepresentation	γ = 1 − ln 2 + ∑_{n=2}^∞ (−1)^n (ζ(n)/n) ⓘ γ = ∑_{n=1}^∞ (1/n − ln((n+1)/n)) ⓘ γ = ∑_{n=2}^∞ (−1)^n ζ(n)/n ⓘ
symbol	γ ⓘ

How these facts were elicited

Referenced by (2)

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

Gamma function → containsConstant → Euler–Mascheroni constant γ ⓘ

Mertens’ theorems → involves → Euler–Mascheroni constant γ ⓘ

subject surface form: Mertens’ third theorem

this entity surface form: Euler–Mascheroni constant

All labels observed (2)

How this entity was disambiguated Show

Statements (48)

How these facts were elicited Show

Referenced by (2)

How this entity was disambiguated

How these facts were elicited