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# Euler–Mascheroni constant

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.

${\displaystyle \gamma :=\lim _{n\to \infty }\left\{\sum _{k=1}^{n}{\frac {1}{k}}-\log n\right\}=\lim _{n\to \infty }\left\{\sum _{k=1}^{n}{\frac {1}{k}}-\int _{1}^{n}{\frac {dx}{x}}\right\}=\lim _{n\to \infty }\left\{\sum _{k=1 \atop \Delta k=1}^{n}{\frac {\Delta k}{k}}-\int _{1}^{n}{\frac {dx}{x}}\right\}=\int _{1}^{\infty }\left({\frac {1}{\lfloor x\rfloor }}-{\frac {1}{x}}\right)\,dx=\int _{1}^{\infty }{\frac {\{x\}}{\lfloor x\rfloor \,x}}\,dx,\,}$
where
 ⌊  x⌋
is the floor function and
 {  x } := x  −  ⌊  x⌋
is the fractional part of
 x
, when
 x   ≥   0
.

Young proved that[1]

${\displaystyle {\frac {1}{2(n+1)}}<\sum _{k=1}^{n}{\frac {1}{k}}-\log n-\gamma <{\frac {1}{2n}},\,}$

hence

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

## Decimal expansion

The decimal expansion of the Euler–Mascheroni constant is

${\displaystyle \gamma =0.57721566490153286060651209008240243104215933593992\ldots ,\,}$
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

${\displaystyle \gamma ={0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}.\,}$
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

${\displaystyle \gamma ^{2}=0.3331779238077186743183761363552442\ldots ,\,}$
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

${\displaystyle {\frac {1}{\gamma }}=1.7324547146006334735830253158608296811557765522668050220484361328706553140865524300883284\ldots ,\,}$
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
${\displaystyle \lim _{s\to 1}{\bigg [}\zeta (s)-{\frac {1}{s-1}}{\bigg ]}=\gamma .\,}$