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 $\scriptstyle \gamma \,$ . Euler's constant $\scriptstyle \gamma \,$ 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 $\scriptstyle \lfloor x \rfloor \,$ is the floor function and $\scriptstyle \{ x \} \,:=\, x - \lfloor x \rfloor \,$ is the fractional part of $\scriptstyle x \,$ , when $\scriptstyle x \,\ge\, 0 \,$ .

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 $\scriptstyle \gamma \,$ is irrational.^[2]^[3]

Decimal expansion

The decimal expansion of the Euler–Mascheroni constant is

$\gamma = 0.57721566490153286060651209008240243104215933593992 \ldots, \,$

which is pretty close to $\scriptstyle \frac{1}{\sqrt 3} \,=\, 0.577350269189626\ldots \,$ ( $\scriptstyle \gamma \,=\, \frac{1}{\sqrt 3} \times 0.999766858534\ldots \,$ )!

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) 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 $\scriptstyle \frac{1}{3} \,=\, 0.3333333333\ldots \,$ ( $\scriptstyle \gamma^2 \,=\, \frac{1}{3} \times 0.999533771423156\ldots \,$ )!

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 $\scriptstyle \sqrt{3} \,=\, 1.732050807568877\ldots \,$ ( $\scriptstyle \frac{1}{\gamma} \,=\, \sqrt{3} \times 1.0002331958335\ldots \,$ )!

A098907 Decimal expansion of 1/gamma, 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 $\scriptstyle s \,=\, 1 \,$ , we obtain

$\lim_{s \to 1} \bigg[ \zeta(s) - \frac{1}{s-1}\bigg] = \gamma. \,$

Notes

↑ 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.
↑ Weisstein, Eric W., Irrational Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/IrrationalNumber.html]
↑ John Albert, Some unsolved problems in number theory, Department of Mathematics, University of Oklahoma.

External links

Weisstein, Eric W., Euler-Mascheroni Constant, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Euler-MascheroniConstant.html]

[0] 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.

[1] Weisstein, Eric W., Irrational Number, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/IrrationalNumber.html]

[2] John Albert, Some unsolved problems in number theory, Department of Mathematics, University of Oklahoma.

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Euler–Mascheroni constant

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Contents

Decimal expansion

Continued fraction expansion

Square of the Euler–Mascheroni constant

Reciprocal

Laurent expansion of the Riemann zeta function

See also

Notes

External links

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