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Euler–Mascheroni constant

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Euler–Mascheroni constant

The area of the blue region converges to the Euler–Mascheroni constant.

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (\gamma).

It is defined as the limiting difference between the harmonic series and the natural logarithm:

\begin{align} \gamma &= \lim_{n \rightarrow \infty } \left( -\ln(n) + \sum_{k=1}^n \frac{1}{k} \right)\\ &=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx. \end{align}

Here, \lfloor x\rfloor represents the floor function.

The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is

0.57721566490153286060651209008240243104215933593992.[1]
Binary 0.1001001111000100011001111110001101111101...
Decimal 0.5772156649015328606065120900824024310421...
Hexadecimal 0.93C467E37DB0C7A4D1BE3F810152CB56A1CECC3A...
Continued fraction [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, … ][2]
(It is not known whether this continued fraction is finite, infinite periodic or infinite non-periodic.
Shown in linear notation)

History

The constant first appeared in a 1734 paper by the [5]

Appearances

The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):

Properties

The number \gamma has not been proved algebraic or transcendental. In fact, it is not even known whether \gamma is irrational. Continued fraction analysis reveals that if \gamma is rational, its denominator must be greater than 10242080.[6] The ubiquity of \gamma revealed by the large number of equations below makes the irrationality of \gamma a major open question in mathematics. Also see Sondow (2003a).

Relation to gamma function

\gamma is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:

\ -\gamma = \Gamma'(1) = \Psi(1).

This is equal to the limits:

-\gamma = \lim_{z\to 0} \left\{\Gamma(z) - \frac1{z} \right\} = \lim_{z\to 0} \left\{\Psi(z) + \frac1{z} \right\}.

Further limit results are (Krämer, 2005):

\lim_{z\to 0} \frac1{z}\left\{\frac1{\Gamma(1+z)} - \frac1{\Gamma(1-z)} \right\} = 2\gamma
\lim_{z\to 0} \frac1{z}\left\{\frac1{\Psi(1-z)} - \frac1{\Psi(1+z)} \right\} = \frac{\pi^2}{3\gamma^2}.

A limit related to the beta function (expressed in terms of gamma functions) is

\gamma = \lim_{n \to \infty} \left \{\frac{ \Gamma(\frac{1}{n}) \Gamma(n+1)\, n^{1+1/n}}{\Gamma(2+n+\frac{1}{n})} - \frac{n^2}{n+1} \right\}
\gamma = \lim\limits_{m \to \infty}\sum_{k=1}^m{m \choose k}\frac{(-1)^k}{k}\ln(\Gamma(k+1)).

Relation to the zeta function

\gamma can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:

\begin{align}\gamma &= \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m} \\ &= \ln \left ( \frac{4}{\pi} \right ) + \sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{2^{m-1}m}.\end{align}

Other series related to the zeta function include:

\begin{align} \gamma &= \frac{3}{2}- \ln 2 - \sum_{m=2}^\infty (-1)^m\,\frac{m-1}{m} [\zeta(m)-1] \\ &= \lim_{n \to \infty} \left [ \frac{2\,n-1}{2\,n} - \ln\,n + \sum_{k=2}^n \left ( \frac{1}{k} - \frac{\zeta(1-k)}{n^k} \right ) \right ] \\ &= \lim_{n \to \infty} \left [ \frac{2^n}{e^{2^n}} \sum_{m=0}^\infty \frac{2^{m \,n}}{(m+1)!} \sum_{t=0}^m \frac{1}{t+1} - n\, \ln 2+ O \left ( \frac{1}{2^n\,e^{2^n}} \right ) \right ].\end{align}

The error term in the last equation is a rapidly decreasing function of n. As a result, the formula is well-suited for efficient computation of the constant to high precision.

Other interesting limits equaling the Euler–Mascheroni constant are the antisymmetric limit (Sondow, 1998)

\gamma = \lim_{s \to 1^+} \sum_{n=1}^\infty \left ( \frac{1}{n^s}-\frac{1}{s^n} \right ) = \lim_{s \to 1} \left ( \zeta(s) - \frac{1}{s-1} \right ) = \lim_{s \to 0} \frac{\zeta(1+s)+\zeta(1-s)}{2}

and de la Vallée-Poussin's formula

\begin{align} \gamma = \lim_{n \to \infty} \frac{1}{n}\, \sum_{k=1}^n \left ( \left \lceil \frac{n}{k} \right \rceil - \frac{n}{k} \right ).\end{align}

Closely related to this is the rational zeta series expression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:

\gamma = \sum_{k=1}^n \frac{1}{k} - \ln n - \sum_{m=2}^\infty \frac{\zeta (m,n+1)}{m}

where ζ(s,k) is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, Hn. Expanding some of the terms in the Hurwitz zeta function gives:

H_n = \ln n + \gamma + \frac {1} {2n} - \frac {1} {12n^2} + \frac {1} {120n^4} - \varepsilon , where 0 < \varepsilon < \frac {1} {252n^6}.

Integrals

\gamma equals the value of a number of definite integrals:

\begin{align}\gamma &= - \int_0^\infty { e^{-x} \ln x }\,dx = -4\int_0^\infty { e^{-x^2} x \ln x }\,dx\\ &= -\int_0^1 \ln\ln\left (\frac{1}{x}\right) dx \\ &= \int_0^\infty \left (\frac1{e^x-1}-\frac1{xe^x} \right)dx = \int_0^1\left(\frac 1{\ln x} + \frac 1{1-x}\right)dx\\ &= \int_0^\infty \left (\frac1{1+x^k}-e^{-x} \right)\frac{dx}{x},\quad k>0\\ &= \int_0^1 H_{x} dx \end{align}

where H_{x} is the fractional Harmonic number.

Definite integrals in which \gamma appears include:

\int_0^\infty { e^{-x^2} \ln x }\,dx = -\tfrac14(\gamma+2 \ln 2) \sqrt{\pi}
\int_0^\infty { e^{-x} \ln^2 x }\,dx = \gamma^2 + \frac{\pi^2}{6} .

One can express \gamma using a special case of Hadjicostas's formula as a double integral (Sondow 2003a, 2005) with equivalent series:

\gamma = \int_{0}^{1}\int_{0}^{1} \frac{x-1}{(1-x\,y)\ln(x\,y)} \, dx\,dy = \sum_{n=1}^\infty \left ( \frac{1}{n}-\ln\frac{n+1}{n} \right ).

An interesting comparison by J. Sondow (2005) is the double integral and alternating series

\ln \left ( \frac{4}{\pi} \right ) = \int_{0}^{1}\int_{0}^{1} \frac{x-1}{(1+x\,y)\ln(x\,y)} \, dx\,dy = \sum_{n=1}^\infty (-1)^{n-1} \left( \frac{1}{n}-\ln\frac{n+1}{n} \right).

It shows that \ln \left ( \frac{4}{\pi} \right ) may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series (see Sondow 2005 #2)

\sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} = \gamma
\sum_{n=1}^\infty \frac{N_1(n) - N_0(n)}{2n(2n+1)} = \ln \left ( \frac{4}{\pi} \right )

where N1(n) and N0(n) are the number of 1's and 0's, respectively, in the base 2 expansion of n.

We have also Catalan's 1875 integral (see Sondow and Zudilin)

\gamma = \int_0^1 \frac{1}{1+x} \sum_{n=1}^\infty x^{2^n-1} \, dx.

Series expansions

Euler showed that the following infinite series approaches \gamma :

\gamma = \sum_{k=1}^\infty \left[ \frac{1}{k} - \ln \left( 1 + \frac{1}{k} \right) \right].

The series for \gamma is equivalent to series Nielsen found in 1897:

\gamma = 1 - \sum_{k=2}^{\infty}(-1)^k\frac{\lfloor\log_2 k\rfloor}{k+1}.

In 1910, Vacca found the closely related series:

{ \gamma = \sum_{k=2}^\infty (-1)^k \frac{ \left \lfloor \log_2 k \right \rfloor}{k} = \frac12-\frac13 + 2\left(\frac14 - \frac15 + \frac16 - \frac17\right) + 3\left(\frac18 - \frac19 + \frac1{10} - \frac1{11} + \dots - \frac1{15}\right) + \dots }

where \log_2 is the logarithm of base 2 and \lfloor \, \rfloor is the floor function.

In 1926 he found a second series:

{\gamma + \zeta(2) = \sum_{k=2}^\infty\left(\frac1{\lfloor \sqrt{k} \rfloor^2} - \frac1{k}\right) = \sum_{k=2}^{\infty} \frac{k - \lfloor\sqrt{k}\rfloor^2}{k\lfloor\sqrt{k}\rfloor^2} = \frac12 + \frac23 + \frac1{2^2} \sum_{k=1}^{2 \times 2} \frac k {k+2^2} + \frac1{3^2} \sum_{k=1}^{3 \times 2} \frac k {k+3^2} + \dots}.

From the Malmsten-Kummer-expansion for the logarithm of the gamma function we get:

\gamma = \ln\pi - 4\ln\Gamma(\tfrac34) + \frac4{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\frac{\ln(2k+1)}{2k+1}.

Series of prime numbers:

\begin{align} \gamma = \lim_{n \to \infty} \left( \ln n - \sum_{p \le n} \frac{ \ln p }{ p-1 } \right)\end{align}.

Series relating to square roots:

\gamma = \lim_{n \rightarrow \infty}\left [ \sum_{k=1}^n \frac{1}{k} - \ln \sqrt { \sum_{k=1}^n k } \right ] - \ln \sqrt 2 [7]

Asymptotic expansions

\gamma equals the following asymptotic formulas (where H_n is the nth harmonic number.)

\gamma \sim H_n - \ln \left( n \right) - \frac{1} + \frac{1} - \frac{1} + ...
(Euler)
\gamma \sim H_n - \ln \left( {n + \frac{1}{2} + \frac{1} - \frac{1} + ...} \right)
(Negoi)
\gamma \sim H_n - \frac{2} - \frac{1} + \frac{1} - ...
(Cesaro)

The third formula is also called the Ramanujan expansion.

Relations with the reciprocal logarithm

The reciprocal logarithm function (Krämer, 2005)

\frac{z}{\ln(1-z)} = \sum_{n=0}^{\infty}C_nz^n, \quad |z|<1,

has a deep connection with Euler's constant and was studied by James Gregory in connection with numerical integration. The coefficients C_n are called Gregory coefficients; the first six were given in a letter to John Collins in 1670. From the equations

C_0 = -1\;,\quad \sum_{k=0}^n\frac{C_k}{n+1-k} = 0,\quad n=1,2,3,\dots
, which can be used recursively to get these coefficients for all n \ge 1, we get the table
n 1 2 3 4 5 6 7 8 9 10 OEIS sequences
Cn \tfrac12 \tfrac1{12} \tfrac1{24} \tfrac{19}{720} \tfrac3{160} \tfrac{863}{60480} \tfrac{275}{24192} \tfrac{33953}{3628800} \tfrac{8183}{1036800} \tfrac{3250433}{479001600} A002206 (numerators),

A002207 (denominators)

Gregory coefficients are similar to Bernoulli numbers and satisfy the asymptotic relation

C_n = \frac1{n\ln^2 n} - \mathcal{O}\left(\frac1{n\ln^3 n}\right),\quad n\to\infty,

and the integral representation

C_n = \int_0^{\infty}\frac{dx}{(1+x)^n\left(\ln^2 x + \pi^2\right)},\quad n=0,1,2,\dots.

Euler's constant has the integral representations

\gamma = \int_0^{\infty}\frac{\ln(1+x)}{\ln^2 x + \pi^2}\cdot\frac{dx}{x^2} = \int_{-\infty}^{\infty}\frac{\ln(1+e^{-x})}{x^2 + \pi^2}\,e^x\,dx.

A very important expansion of Gregorio Fontana (1780) is:

\begin{align} H_n &= \gamma + \log n + \frac1{2n} - \sum_{k=2}^{\infty}\frac{(k-1)!C_k}{n(n+1)\dots(n+k-1)},\quad n=1,2,\dots,\\ &= \gamma + \log n + \frac1{2n} - \frac1{12n(n+1)} - \frac1{12n(n+1)(n+2)} - \frac{19}{120n(n+1)(n+2)(n+3)} - \dots \end{align}

which is convergent for all n.

Weighted sums of the Gregory coefficients give different constants:

\begin{align} 1 &= \sum_{n=1}^{\infty}C_n = \tfrac12 + \tfrac1{12} + \tfrac1{24} + \tfrac{19}{720} + \tfrac3{160} + \dots,\\ \frac1{\log2} - 1 &= \sum_{n=1}^{\infty}(-1)^{n+1}C_n = \tfrac12 - \tfrac1{12} + \tfrac1{24} - \tfrac{19}{720} + \tfrac3{160} - \dots,\\ \gamma &= \sum_{n=1}^{\infty}\frac{C_n}{n} = \tfrac12 + \tfrac1{24} + \tfrac1{72} + \tfrac{19}{2880} + \tfrac3{800} + \dots. \end{align}

eγ

The constant eγ is important in number theory. Some authors denote this quantity simply as \gamma^\prime . eγ equals the following limit, where pn is the nth prime number:

e^\gamma = \lim_{n \to \infty} \frac {1} {\ln p_n} \prod_{i=1}^n \frac {p_i} {p_i - 1}.

This restates the third of Mertens' theorems.[8] The numerical value of eγ is:

1.78107241799019798523650410310717954916964521430343 … A073004.

Other infinite products relating to eγ include:

\frac{e^{1+\gamma /2}}{\sqrt{2\,\pi}} = \prod_{n=1}^\infty e^{-1+1/(2\,n)}\,\left (1+\frac{1}{n} \right )^n
\frac{e^{3+2\gamma}}{2\, \pi} = \prod_{n=1}^\infty e^{-2+2/n}\,\left (1+\frac{2}{n} \right )^n.

These products result from the Barnes G-function.

We also have

e^{\gamma} = \left ( \frac{2}{1} \right )^{1/2} \left (\frac{2^2}{1 \cdot 3} \right )^{1/3} \left (\frac{2^3 \cdot 4}{1 \cdot 3^3} \right )^{1/4} \left (\frac{2^4 \cdot 4^4}{1 \cdot 3^6 \cdot 5} \right )^{1/5} \cdots

where the nth factor is the (n+1)st root of

\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}.

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.

Continued fraction

The continued fraction expansion of \gamma is of the form [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] A002852, of which there is no apparent pattern. The continued fraction has at least 470,000 terms,[6] and it has infinitely many terms if and only if \gamma is irrational.

Generalizations

abm(x) = \gamma_ {-x}

Euler's generalized constants are given by

\gamma_\alpha = \lim_{n \to \infty} \left[ \sum_{k=1}^n \frac{1}{k^\alpha} - \int_1^n \frac{1}{x^\alpha} \, dx \right],

for 0 < α < 1, with \gamma as the special case α = 1.[9] This can be further generalized to

c_f = \lim_{n \to \infty} \left[ \sum_{k=1}^n f(k) - \int_1^n f(x) \, dx \right]

for some arbitrary decreasing function f. For example,

f_n(x) = \frac{\ln^n x}{x}

gives rise to the Stieltjes constants, and

f_a(x) = x^{-a}

gives

\gamma_{f_a} = \frac{(a-1)\zeta(a)-1}{a-1}

where again the limit

\gamma = \lim_{a\to1}\left[ \zeta(a) - \frac{1}{a-1}\right]

appears.

A two-dimensional limit generalization is the Masser–Gramain constant.

Euler-Lehmer constants are given by summation of inverses of numbers in a common modulo class[10] ,[11]

\gamma(a,q) = \lim_{x\to \infty}\left ( \sum_{0

The basic properties are

\gamma(0,q) = \frac{\gamma -\log q}{q},
\sum_{a=0}^{q-1} \gamma(a,q)=\gamma,
q\gamma(a,q) = \gamma-\sum_{j=1}^{q-1}e^{-2\pi aij/q}\log(1-e^{2\pi ij/q}),

and if gcd(a,q)=d then

q\gamma(a,q) = \frac{q}{d}\gamma(a/d,q/d)-\log d.

Published digits

Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 20th–22nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.

Published Decimal Expansions of \gamma
Date Decimal digits Author
1734 5 Leonhard Euler
1735 15 Leonhard Euler
1790 19 Lorenzo Mascheroni
1809 22 Johann G. von Soldner
1811 22 Carl Friedrich Gauss
1812 40 Friedrich Bernhard Gottfried Nicolai
1857 34 Christian Fredrik Lindman
1861 41 Ludwig Oettinger
1867 49 William Shanks
1871 99 James W.L. Glaisher
1871 101 William Shanks
1877 262 J. C. Adams
1952 328 John William Wrench, Jr.
1961 1050 Helmut Fischer and Karl Zeller
1962 1,271 Donald Knuth
1962 3,566 Dura W. Sweeney
1973 4,879 William A. Beyer and Michael S. Waterman
1977 20,700 Richard P. Brent
1980 30,100 Richard P. Brent & Edwin M. McMillan
1993 172,000 Jonathan Borwein
2009 29,844,489,545 Alexander J. Yee & Raymond Chan[12]
2013 119,377,958,182 Alexander J. Yee[12]

See also

Notes

Footnotes
  1. ^ A001620
  2. ^ A002852
  3. ^
  4. ^ Carl Anton Bretschneider: Theoriae logarithmi integralis lineamenta nova (13 October 1835), Journal für die reine und angewandte Mathematik 17, 1837, pp. 257–285 (in Latin; "γ = c = 0,577215 664901 532860 618112 090082 3.." on [Euler–Mascheroni constant p. 260])
  5. ^ γ" on p. 578)
  6. ^ a b Havil 2003 p 97.
  7. ^ http://mathworld.wolfram.com/Euler-MascheroniConstant.html
  8. ^ http://mathworld.wolfram.com/MertensConstant.html (14)
  9. ^ Havil, 117-118
  10. ^
  11. ^
  12. ^ a b Nagisa – Large Computations
References
  • PDF
  • Derives γ as sums over Riemann zeta functions.
  • Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, γ."
  • Gourdon, Xavier, and Sebah, P. (2004) "The Euler constant: γ."
  • Donald Knuth (1997) The Art of Computer Programming, Vol. 1, 3rd ed. Addison-Wesley. ISBN 0-201-89683-4
  • Krämer, Stefan (2005) Die Eulersche Konstante γ und verwandte Zahlen. Diplomarbeit, Universität Göttingen.
  • with an Appendix by Sergey Zlobin
  • Ramanujan Journal 12: 225-244.
  • James Whitbread Lee Glaisher (1872), "On the history of Euler's constant". Messenger of Mathematics vol.1, p. 25-30, JFM 03.0130.01
  • (submitted 1835)
  • Lorenzo Mascheroni (1790). "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur". Galeati, Ticini.
  • Lorenzo Mascheroni (1792). "Adnotationes ad calculum integralem Euleri. In quibus nonnullae formulae ab Eulero propositae evolvuntur". Galeati, Ticini. Both online at Euler–Mascheroni constant at Google Books

External links

  • Weisstein, Eric W., "Euler-Mascheroni constant", MathWorld.
  • Jonathan Sondow.
  • Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
  • Further formulae which make use of the constant: Gourdon and Sebah (2004).
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