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Euler–Mascheroni constant

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The Euler–Mascheroni constant (also called Euler's constant), named after Leonhard Euler and Lorenzo Mascheroni, is a constant occurring in analysis and number theory, usually denoted by the lowercase Greek letter
γ
. Euler's constant
γ
should not be confused with the base
e
of the natural logarithm, which is sometimes called Euler's number.

It is defined as the limiting difference between the harmonic series and the natural logarithm, i.e.

\gamma := \lim_{n \to \infty} \left\{ \sum_{k=1}^{n} \frac{1}{k} - \log n \right\}
 = \lim_{n \to \infty} \left\{ \sum_{k=1}^{n} \frac{1}{k} - \int_{1}^{n} \frac{dx}{x} \right\}
 = \lim_{n \to \infty} \left\{ \sum_{k=1 \atop \Delta k = 1}^{n} \frac{\Delta k}{k} - \int_{1}^{n} \frac{dx}{x} \right\}
 = \int_{1}^{\infty} \left( \frac{1}{\lfloor x \rfloor} - \frac{1}{x} \right) \, dx
 = \int_{1}^{\infty} \frac{\{ x \}}{\lfloor x \rfloor \, x} \, dx, \,
where
⌊ x
is the floor function and
{  x } := x
⌊ x
is the fractional part of
x
, when
x ≥ 0
.

Young proved that[1]

\frac{1}{2(n + 1)} < \sum_{k=1}^{n} \frac{1}{k} - \log n - \gamma < \frac{1}{2n}, \,

hence

\sum_{k=1}^{n} \frac{1}{k} - \log n - \gamma \sim \frac{1}{2n}. \,
It is not known whether
γ
is irrational.[2][3]

Contents

Decimal expansion

The decimal expansion of the Euler–Mascheroni constant is

\gamma = 0.57721566490153286060651209008240243104215933593992 \ldots, \,
which is pretty close to
1
2  3 
  = 0.577350269189626
(
γ =
1
2  3 
  × 0.999766858534
)!

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

\gamma = {0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{\ddots}}}}}}}}. \,
A002852 Continued fraction for Euler's constant (or Euler–Mascheroni constant)
γ
(gamma).
{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

\gamma^2 = 0.3331779238077186743183761363552442 \ldots, \,
which is pretty close to
1
3
  = 0.3333333333
(
γ 2 =
1
3
  × 0.999533771423156
)!

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

\frac{1}{\gamma} = 1.7324547146006334735830253158608296811557765522668050220484361328706553140865524300883284 \ldots, \,
which is pretty close to
2  3  = 1.732050807568877
(
1 ⧸ γ = 2  3  × 1.0002331958335
)!
A098907 Decimal expansion of
1 ⧸ γ
, where gamma is Euler–Mascheroni constant.
{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 about
s = 1
, we obtain
\lim_{s \to 1} \bigg[ \zeta(s) - \frac{1}{s-1}\bigg] = \gamma. \,

See also

Notes

  1. Štefan Porubský: Euler-Mascheroni 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, web-page http://www.cs.cas.cz/portal/AlgoMath/MathematicalAnalysis/MathematicalConstants/EulerMascheroni.htm.
  2. Weisstein, Eric W., Irrational Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/IrrationalNumber.html]
  3. John Albert, Some unsolved problems in number theory, Department of Mathematics, University of Oklahoma.

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