Euler-Mascheroni Constant

The Euler-Mascheroni constant is denoted $\gamma$ (or sometimes $C$) and has the numerical value

$$\gamma\approx 0.577215664901532860606512090082402431042\ldots$$ (1)

(Sloane's A001620). The Continued Fraction of the Euler-Mascheroni constant is [0, 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (Sloane's A002852). The first few Convergents are 1, 1/2, 3/5, 4/7, 11/19, 15/26, 71/123, 228/395, 3035/5258, 15403/26685, ... (Sloane's A046114 and A046115). The positions at which the digits 1, 2, ... first occur in the Continued Fraction are 2, 4, 9, 8, 11, 69, 24, 14, 139, 52, 22, ... (Sloane's A033149). The sequence of largest terms in the Continued Fraction is 1, 2, 4, 13, 40, 49, 65, 399, 2076, ... (Sloane's A033091), which occur at positions 2, 4, 8, 10, 20, 31, 34, 40, 529, ... (Sloane's A033092).

It is not known if this constant is Irrational, let alone Transcendental. However, Conway and Guy (1996) are ``prepared to bet that it is transcendental,'' although they do not expect a proof to be achieved within their lifetimes.

The Euler-Mascheroni constant arises in many integrals

$$\gamma \equiv - \int^\infty_0 e^{-x}\ln x\,dx$$ (2)

$$= \int_0^\infty \left({{1\over 1-e^{-x}} - {1\over x}}\right)e^{-x}\,dx$$ (3)

$$= \int_0^\infty {1\over x}\left({{1\over 1+x} - e^{-x}}\right)\, dx.$$ (4)

and sums

$$\gamma \equiv 1 + \sum_{k=2}^\infty \left[{{1\over k} + \ln\left({k-1\over k}\right)}\right]$$ (5)

$$= \lim_{m\to \infty} \left({\sum_{n=1}^m {1\over n} - \ln m}\right)$$ (6)

$$= \sum_{n=2}^\infty (-1)^n {\zeta(n)\over n}$$ (7)

$$= \ln\left({4\over\pi}\right)- \sum_{n=1}^\infty{(-1)^n\zeta(n+1)\over 2^n(n+1)}$$ (8)

$$= \lim_{n\to\infty}\left[{\,\sum_{k=1}^n k^{-1} - \ln n - {1\over 2n} + \sum_{k=1}^n {B_{2k}\over (2k)n^{2k}}}\right],$$ (9)

where $\zeta(z)$ is the Riemann Zeta Function and $B_n$ are the Bernoulli Numbers. It is also given by the Euler Product

$$e^\gamma=\lim_{n\to\infty} {1\over\ln n} \prod_{i=1}^n {1\over 1-{1\over p_i}},$$ (10)

where the product is over Primes $p$. Another connection with the Primes was provided by Dirichlet's 1838 proof that the average number of Divisors of all numbers from 1 to $n$ is asymptotic to

$${\sum_{i=1}^n \sigma_0(i)\over n}\sim\ln n+2\gamma-1$$ (11)

(Conway and Guy 1996). de la Vallée Poussin (1898) proved that, if a large number $n$ is divided by all Primes $\leq n$, then the average amount by which the Quotient is less than the next whole number is $\gamma$.

Infinite Products involving $\gamma$ also arise from the G-Function with Positive Integer $n$. The cases $G(2)$ and $G(3)$ give

$$\prod_{n=1}^\infty e^{-1+1/(2n)}\left({1+{1\over n}}\right)^n = {e^{1+\gamma/2}\over\sqrt{2\pi}}$$ (12)

$$\prod_{n=1}^\infty e^{-2+2/n}\left({1+{2\over n}}\right)^n = {e^{3+2\gamma}\over 2\pi}.$$ (13)

The Euler-Mascheroni constant is also given by the limits

$$\gamma = \lim_{s\to 1}{\zeta(s)-1\over s-1}$$ (14)

$$= -\Gamma'(1)$$ (15)

$$= \lim_{x\to\infty}\left[{x-\Gamma\left({1\over x}\right)}\right]$$ (16)

(Le Lionnais 1983).

The difference between the $n$th convergent in (6) and $\gamma$ is given by

$$\sum_{k=1}^n {1\over k}-\ln n-\gamma=\int_n^\infty {x-\left\lfloor{x}\right\rfloor \over x^2}\,dx,$$ (17)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function, and satisfies the Inequality

$${1\over 2(n+1)}<\sum_{k=1}^n {1\over k}-\ln n-\gamma<{1\over 2n}$$ (18)

(Young 1991). A series with accelerated convergence is

$$\gamma={\textstyle{3\over 2}}-\ln 2-\sum_{m=2}^\infty (-1)^m{m-1\over m}[\zeta(m)-1]$$ (19)

(Flajolet and Vardi 1996). Another series is

$$\gamma=\sum_{n=1}^\infty (-1)^n {\left\lfloor{\lg n}\right\rfloor \over n}$$ (20)

(Vacca 1910, Gerst 1969), where Lg is the Logarithm to base 2. The convergence of this series can be greatly improved using Euler's Convergence Improvement transformation to

$$\gamma=\sum_{k=1}^\infty 2^{-(k+1)} \sum_{j=0}^{k-1} {1\over{2^{k-j}+j\choose j}},$$ (21)

where ${a\choose b}$ is a Binomial Coefficient (Beeler et al. 1972, Item 120, with $k-j$ replacing the undefined $i$). Bailey (1988) gives

$$\gamma={2^n\over e^{2^n}}\sum_{m=0}^\infty {2^{mn}\over(m+1)... ...ver t+1}-n\ln 2+{\mathcal O}\left({1\over 2^n e^{2^n}}\right),$$ (22)

which is an improvement over Sweeney (1963).

The symbol $\gamma$ is sometimes also used for

$$\gamma'\equiv e^\gamma \approx 1.781072$$ (23)

(Gradshteyn and Ryzhik 1979, p. xxvii).

Odena (1982-1983) gave the strange approximation

$$(0.11111111)^{1/4}=0.577350\ldots,$$ (24)

and Castellanos (1988) gave

$$({\textstyle{7\over 83}})^{2/9} = 0.57721521\ldots$$ (25)

$$\left({520^2+22\over 52^4}\right)^{1/6} = 0.5772156634\ldots$$ (26)

$$\left({80^3+92\over 61^4}\right)^{1/6} = 0.57721566457\ldots$$ (27)

$${990^3-55^3-79^2-4^2\over 70^5} = 0.5772156649015295\ldots.$$ (28)

No quadratically converging algorithm for computing $\gamma$ is known (Bailey 1988). 7,000,000 digits of $\gamma$ have been computed as of Feb. 1998 (Plouffe).

See also Euler Product, Mertens Theorem, Stieltjes Constants

References

Bailey, D. H. ``Numerical Results on the Transcendence of Constants Involving $\pi$, $e$, and Euler's Constant.'' Math. Comput. 50, 275-281, 1988.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Brent, R. P. ``Computation of the Regular Continued Fraction for Euler's Constant.'' Math. Comput. 31, 771-777, 1977.

Brent, R. P. and McMillan, E. M. ``Some New Algorithms for High-Precision Computation of Euler's Constant.'' Math. Comput. 34, 305-312, 1980.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988.

Conway, J. H. and Guy, R. K. ``The Euler-Mascheroni Number.'' In The Book of Numbers. New York: Springer-Verlag, pp. 260-261, 1996.

de la Vallée Poussin, C.-J. Untitled communication. Annales de la Soc. Sci. Bruxelles 22, 84-90, 1898.

DeTemple, D. W. ``A Quicker Convergence to Euler's Constant.'' Amer. Math. Monthly 100, 468-470, 1993.

Dirichlet, G. L. J. für Math. 18, 273, 1838.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/euler/euler.html

Flajolet, P. and Vardi, I. ``Zeta Function Expansions of Classical Constants.'' Unpublished manuscript, 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps.

Gerst, I. ``Some Series for Euler's Constant.'' Amer. Math. Monthly 76, 273-275, 1969.

Glaisher, J. W. L. ``On the History of Euler's Constant.'' Messenger of Math. 1, 25-30, 1872.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.

Knuth, D. E. ``Euler's Constant to 1271 Places.'' Math. Comput. 16, 275-281, 1962.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 28, 1983.

Plouffe, S. ``Plouffe's Inverter: Table of Current Records for the Computation of Constants.'' http://www.lacim.uqam.ca/pi/records.html.

Sloane, N. J. A. Sequences A033091, A033092, A033149, A046114, A046114, A001620/M3755, and A002852/M0097 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Sweeney, D. W. ``On the Computation of Euler's Constant.'' Math. Comput. 17, 170-178, 1963.

Vacca, G. ``A New Series for the Eulerian Constant.'' Quart. J. Pure Appl. Math. 41, 363-368, 1910.

Young, R. M. ``Euler's Constant.'' Math. Gaz. 75, 187-190, 1991.