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

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 ⓘ

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