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Euler–Mascheroni constant
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(Redirected from Harmonic series)
γ 
γ 
e 
It is defined as the limiting difference between the harmonic series and the natural logarithm, i.e.
⌊ x⌋ 
{ x } := x − ⌊ x⌋ 
x 
x ≥ 0 
Young proved that^{[1]}
hence
γ 
Contents
Decimal expansion
The decimal expansion of the Euler–Mascheroni constant is

γ =

A001620 Decimal expansion of Euler's constant (or Euler–Mascheroni constant) gamma.

{5, 7, 7, 2, 1, 5, 6, 6, 4, 9, 0, 1, 5, 3, 2, 8, 6, 0, 6, 0, 6, 5, 1, 2, 0, 9, 0, 0, 8, 2, 4, 0, 2, 4, 3, 1, 0, 4, 2, 1, 5, 9, 3, 3, 5, 9, 3, 9, 9, 2, 3, 5, 9, 8, 8, 0, 5, 7, 6, 7, 2, 3, 4, 8, 8, 4, 8, 6, 7, 7, 2, 6, 7, 7, 7, 6, 6, 4, ...}
The value is available in Pari/GP as "Euler" and WolframAlpha as "EulerGamma".
Continued fraction expansion
The simple continued fraction expansion of the Euler–Mascheroni constant is
γ 

{0, 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, 11, 3, 7, 1, 7, 1, 1, 5, 1, 49, 4, 1, 65, 1, 4, 7, 11, 1, 399, 2, 1, 3, 2, 1, 2, 1, 5, 3, 2, 1, 10, 1, 1, 1, 1, 2, 1, 1, 3, 1, 4, 1, 1, 2, 5, 1, 3, 6, 2, 1, 2, 1, 1, ...}
Square of the Euler–Mascheroni constant
The decimal expansion of the square of the Euler–Mascheroni constant is

γ 2 =

A155969 Decimal expansion of the square of the Euler–Mascheroni constant.

{3, 3, 3, 1, 7, 7, 9, 2, 3, 8, 0, 7, 7, 1, 8, 6, 7, 4, 3, 1, 8, 3, 7, 6, 1, 3, 6, 3, 5, 5, 2, 4, 4, 2, 2, 6, 6, 5, 9, 4, 1, 7, 1, 4, 0, 2, 4, 9, 6, 2, 9, 7, 4, 3, 1, 5, 0, 8, 3, 3, 3, 3, 8, 0, 0, 2, 2, 6, 5, 7, 9, 3, 6, 9, 5, 7, 5, 6, 6, ...}
Reciprocal
The decimal expansion of the reciprocal of the Euler–Mascheroni constant is
√ 3 = 1.732050807568877… 
1 / γ = √ 3 × 1.0002331958335… 
A098907 Decimal expansion of
1 / γ 

{1, 7, 3, 2, 4, 5, 4, 7, 1, 4, 6, 0, 0, 6, 3, 3, 4, 7, 3, 5, 8, 3, 0, 2, 5, 3, 1, 5, 8, 6, 0, 8, 2, 9, 6, 8, 1, 1, 5, 5, 7, 7, 6, 5, 5, 2, 2, 6, 6, 8, 0, 5, 0, 2, 2, 0, 4, 8, 4, 3, 6, 1, 3, 2, 8, 7, 0, 6, 5, 5, 3, 1, 4, 0, 8, 6, 5, 5, 2, ...}
Laurent expansion of the Riemann zeta function
From the Laurent expansion of the Riemann zeta function abouts = 1 
See also
 OEIS format for decimal representation of constants
 Stieltjes constants (sometimes referred to as generalized Euler constants)
 Meissel–Mertens constant (the analogue of Euler–Mascheroni constant for the harmonic series of the primes)
Notes
 ↑ Štefan Porubský: EulerMascheroni Constant. Retrieved 2012/9/20 from Interactive Information Portal for Algorithmic Mathematics, Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, webpage http://www.cs.cas.cz/portal/AlgoMath/MathematicalAnalysis/MathematicalConstants/EulerMascheroni.htm.
 ↑ Weisstein, Eric W., Irrational Number, from MathWorld—A Wolfram Web Resource.
 ↑ John Albert, Some unsolved problems in number theory, Department of Mathematics, University of Oklahoma.
External links
 Weisstein, Eric W., EulerMascheroni Constant, from MathWorld—A Wolfram Web Resource.