Euler-Mascheroni constant

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The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory. It is defined as the limiting difference between the harmonic series and the natural logarithm:

$\gamma = \lim_{n \rightarrow \infty } \left( \left( \sum_{k=1}^n \frac{1}{k} \right) - \log (n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx$

It is usually denoted by the lowercase Greek letter γ (gamma). Sometimes it is called simply the Euler constant, though it ought not to be confused with e, which is often called Euler's number.

Its approximate value is 0.57721 56649 01532 86060 65120 90082 40243 10421 59335.

1 History
2 Properties
3 Relations to special functions
4 Asymptotic expansions
5 e to the power of γ
6 Generalizations
7 Cultural appearances
8 Appearances
9 References

The constant was first defined by Swiss mathematician Leonhard Euler in a paper De Progressionibus harmonicus observationes published in 1735. Euler used the notation C for the constant, and initially calculated its value to 6 decimal places. In 1761 he extended this calculation, publishing a value to 16 decimal places. In 1790 Italian mathematician Lorenzo Mascheroni introduced the notation γ for the constant, and attempted to extend Euler's calculation still further, to 32 decimal places, although subsequent calculations showed that he had made errors in the 20th-22nd, 31st and 32nd decimal places. (From the 20th digit, Mascheroni calculated 1811209008239.)

It is not known whether γ is a rational number or not. However, continued fraction analysis shows that if γ is rational, its denominator must be greater than 10²⁴²⁰⁸⁰ (Havil, page 97).

In December 2006, Alex Yee, an undergraduate at Northwestern University, set a record for calculating the Euler-Mascheroni constant to over 116.6 million decimal places.^[1]

As first discovered by Euler, the limit definition can be restated as an explicit series

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

The constant is given by several integrals:

$\gamma = - \int_0^\infty { e^{-x} \log x }\,dx$

$= - \int_0^1 { \log\log\left (\frac{1}{x}\right ) }\,dx$

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

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

Other integrals that include $γ$ are:

$\int_0^\infty { e^{-x^2} \log(x) }\,dx = -1/4(\gamma+2 \log 2) \sqrt{\pi}$

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

One can express $γ$ also as a double integral (Sondow 2003, 2005) with equivalent series:

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

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

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

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

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

$\sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} = \gamma$

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

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

The series for $γ$ is equivalent to Vacca's interesting 1910 sum

$\gamma = \sum_{m=1}^\infty (-1)^m \frac{ \left \lfloor \log_2 m \right \rfloor}{m}$

where $log 2$ is the logarithm to the base 2 and $\left \lfloor \, \right \rfloor$ is the floor function.

Vacca's series may be obtained by manipulation of Catalan's 1875 integral (see Sondow and Zudilin)

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

$γ$ can also be expressed as an infinite sum with terms involving the values of the Riemann zeta function at positive integers:

$\gamma = \sum_{m=2}^{\infty} \frac{(-1)^m\zeta(m)}{m}$

$= \log \left ( \frac{4}{\pi} \right ) + \sum_{m=1}^{\infty} \frac{(-1)^{m-1} \zeta(m+1)}{2^m (m+1)}.$

Other Zeta-related series include

$\gamma = \frac{3}{2}- \log 2 - \sum_{m=2}^\infty (-1)^m\,\frac{m-1}{m} [\zeta(m)-1]$

$= \lim_{n \to \infty} \left [ \frac{2\,n-1}{2\,n} - \log\,n + \sum_{k=2}^n \left ( \frac{1}{k} - \frac{\zeta(1-k)}{n^k} \right ) \right ].$

$= \lim_{n \to \infty} \left [ \frac{2^n}{e^{2^n}} \sum_{m=0}^\infty \frac{2^{m \,n}}{(m+1)!} \sum_{t=0}^m \frac{1}{t+1} - n\, \log 2+ O \left ( \frac{1}{2^n\,e^{2^n}} \right ) \right ]$

The error term in last identity is a rapidly decreasing function of n. As a result, the formula is well-suited to efficiently computing the constant to high precision.

A limit related to the Beta function (in terms of Gamma functions) is

$\gamma = \lim_{n \to \infty} \left [ \frac{ \Gamma(\frac{1}{n}) \Gamma(n+1)\, n^{1+1/n}}{\Gamma(2+n+\frac{1}{n})} - \frac{n^2}{n+1} \right ].$

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

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

and

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

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

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

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

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

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

The constant can also be calculated as a derivative of Euler's Gamma function:

γ = - Γ'(1).

Asymptotic formulas for Euler's gamma are given by (Where $H n$ is the nth harmonic number.)

$\gamma \sim H_n - \ln \left( n \right) - \frac{1}{{2n}} + \frac{1}{{12n^2 }} - \frac{1}{{120n^4 }} + ...$

(Euler)

$\gamma \sim H_n - \ln \left( {n + \frac{1}{2} + \frac{1}{{24n}} - \frac{1}{{48n^3 }} + ...} \right)$

(Negoi)

$\gamma \sim H_n - \frac{{\ln \left( n \right) + \ln \left( {n + 1} \right)}}{2} - \frac{1}{{6n\left( {n + 1} \right)}} + \frac{1}{{30n^2 \left( {n + 1} \right)^2 }} - ...$

(Cesaro)

The third formula is also called the Ramanujan expansion.

The constant e^γ is also important in number theory. Occasionally, e^γ is denoted $y'$ It is expressed with the following limit, where p_n is the n-th prime number:

$e^\gamma = \lim_{n \to \infty} \frac {1} {\log p_n} \prod_{i=1}^n \frac {p_i} {p_i - 1} ,$

which is a restatement of the third of Mertens' theorems. The numerical value of e^γ is:

$e^\gamma =1.78107241799019798523650410310717954916964521430343\dots$

Other infinite products relating to e^γ include

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

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

Both of 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 is due to J. Ser (1926). It was rediscovered by J. Sondow (2003) using hypergeometric functions.

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 γ as the special case α = 1.^[2] 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,

$f_n(x) = \frac{\log^n x}{x}$

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.

The Euler-Mascheroni constant was used in the solution to the Car Talk puzzler for the week of 23 October 2006. The question involves a car which repeatedly slows down as it nears its destination. The answer presented on the popular radio show uses the Euler-Mascheroni constant.

The Euler-Mascheroni constant appears, among other places, in:

an inequality for Euler's totient function
the growth rate of the divisor function
a product formula for the gamma function
calculations of the digamma function
calculation of the Meissel-Mertens constant
expressions involving the exponential integral
the first term of the Taylor series expansion for the Riemann zeta function, where it is the first of the Stieltjes constants
the third of Mertens' theorems.
the Laplace transform of the natural logarithm

^ Wieczner, Jen. "Student Sets World Record For Math Constant Calculation", The Daily Northwestern, Students Publishing Company, 2007-03-02. ISSN 1523-5033. Retrieved on March 16, 2007. (in English)
^ Havil, 117-118

Jonathan M. Borwein, David M. Bradley, Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function". Journal of Computational and Applied Mathematics 121: p.11. (Provides a derivation of the sums over Riemann zeta)

Knuth, Donald E., The Art of Computer Programming, volume 1, Addison-Wesley. 1997 (third edition). ISBN 0-201-89683-4

Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 0-691-09983-9.

Eric W. Weisstein, Euler-Mascheroni constant at MathWorld.
Euler-Mascheroni Constant from the Mathcad Library
Simon Plouffe, Value of γ to 10 million decimal places
Jonathan Sondow, "An antisymmetric formula for Euler's constant," Mathematics Magazine 71 (1998) 219-220.
Jonathan Sondow, "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant" (2002, preprint) with an Appendix by Sergey Zlobin.
Jonathan Sondow, "An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma" (2003, preprint).
Jonathan Sondow, "Criteria for irrationality of Euler's constant," Proceedings of the American Mathematical Society 131 (2003) 3335-3344.
Jonathan Sondow, "Double integrals for Euler's constant and ln 4/pi and an analog of Hadjicostas's formula," American Mathematical Monthly 112 (2005) 61-65.
Jonathan Sondow, "New Vacca-type rational series for Euler's constant and its 'alternating' analog ln 4/pi" (2005, preprint).
Jonathan Sondow and Wadim Zudilin, "Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper," Ramanujan J. (to appear).
Stefan Krämer. Euler's Constant γ=0.577... Its Mathematics and History

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