Euler–Mascheroni constant

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The Euler–Mascheroni constant (also called the Euler constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter γ (gamma). It is defined as the limiting difference between the harmonic series and the natural logarithm:

<math>\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.</math> Its numerical value to 50 decimal places is 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 ... γ should not to be confused with e, known as Euler's number, the base of the natural logarithms.

List of numbers γ - ζ(3) - √2 - √3 - √5 - φ - α - e - π - δ
Binary	0.1001001111000100011...
Decimal	0.5772156649015328606...
Hexadecimal	0.93C467E37DB0C7A4D1B...
Continued fraction	<math>0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{\ \ddots\ {}}}}}}</math> Note that this continued fraction is not periodic.

1 History
2 Appearances
3 Properties
4 Generalizations
5 Known digits
6 Cultural appearances
7 Notes
8 References
9 External links

History

The constant first appeared in a 1735 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notation C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni introduced the notation A for the constant, The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time because of the constant's connection to the gamma function.^[1]

Appearances

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

Expressions involving the exponential integral
The Laplace transform of the natural logarithm
The first term of the Taylor series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants*
Calculations of the digamma function
A product formula for the gamma function
An inequality for Euler's totient function
The growth rate of the divisor function
The calculation of the Meissel-Mertens constant
The third of Mertens' theorems*.
Solution of the second kind to Bessel's equation.

For more information of this nature, see Gourdon and Sebah (2004).

Properties

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Continued fraction analysis^[2] reveals that if γ is rational, its denominator must be greater than 10²⁴²⁰⁸⁰. The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics. Also see Sondow (2003a). For more equations of the sort shown below, see Gourdon and Sebah (2002).

Relation to gamma function

A result of great simplicity relates γ to minus the digamma function Ψ, and hence minus the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:

<math> \ -\gamma = \Gamma'(1) = \Psi(1). </math>

γ is also the limit:

<math> \gamma = \lim_{x \to \infty} \left [ x - \Gamma \left ( \frac{1}{x} \right ) \right ]. </math>

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

<math> \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 ]. </math>

Relation to the zeta function

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

<math>\gamma = \sum_{m=2}^{\infty} \frac{(-1)^m\zeta(m)}{m} </math>

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

Other series related to the zeta function include:

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

<math> = \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 ]. </math>

<math> = \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 ] </math>

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)

<math> \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 ) </math>

and

<math> \gamma = \lim_{x \to \infty} \left [ x - \Gamma \left ( \frac{1}{x} \right ) \right ] </math>

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

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:

<math>\gamma = \sum_{k=1}^n \frac{1}{k} - \log(n) -

\sum_{m=2}^\infty \frac{\zeta (m,n+1)}{m}</math> where <math>\zeta(s,k)</math> 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:

<math>

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

Integrals

γ equals the value of a number of definite integrals:

<math>\gamma = - \int_0^\infty { e^{-x} \log x }\,dx </math>

<math> = - \int_0^1 { \log\log\left (\frac{1}{x}\right ) }\,dx </math>

<math> = \int_0^\infty {\left (\frac{1}{1-e^{-x}}-\frac{1}{x} \right )e^{-x} }\,dx </math>

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

Indefinite integrals in which <math> \gamma </math> appears include:

<math> \int_0^\infty { e^{-x^2} \log x }\,dx = -\tfrac14(\gamma+2 \log 2) \sqrt{\pi} </math>

<math> \int_0^\infty { e^{-x} \log^2 x }\,dx = \gamma^2 + \frac{\pi^2}{6} .</math>

One can express <math> \gamma </math> also as a double integral (Sondow 2003, 2005) with equivalent series:

<math> \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\frac{n+1}{n} \right ).

</math>

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

<math> \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\frac{n+1}{n} \right). </math>

It shows that <math>\log \left ( \frac{4}{\pi} \right )</math> may be thought of as an "alternating Euler constant". The two constants are also related by the pair of series (see Sondow 2005 #2)

<math> \sum_{n=1}^\infty \frac{N_1(n) + N_0(n)}{2n(2n+1)} = \gamma </math>

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

where <math> N_1(n) </math> and <math> N_0(n) </math> are the number of 1's and 0's, respectively, in the base 2 expansion of <math> n </math>.

Series expansions

Euler showed that the following infinite series approaches γ:

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

The series for <math> \gamma </math> is equivalent to series Nielsen found in 1897: <math> \gamma = 1 - \sum_{k=2}^{\infty}(-1)^k\frac{\lfloor\log_2 k\rfloor}{k+1} </math>.
In 1910, Vacca found the closely related series:

<math>

\gamma = \sum_{k=2}^\infty (-1)^k \frac{ \left \lfloor \log_2 k \right \rfloor}{k}
 = \tfrac12-\tfrac13
 + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right)
 + 3\left(\tfrac18 - \dots - \tfrac1{15}\right) + \dots

</math> where <math> \log_2 </math> is the logarithm to the base 2 and <math> \left \lfloor \, \right \rfloor </math> is the floor function.
In 1926, Vacca found: <math>

\gamma + \zeta(2) = \sum_{k=1}^{\infty} \frac1{k\lfloor\sqrt{k}\rfloor^2}
 = 1 + \tfrac12 + \tfrac13 + \tfrac14\left(\tfrac14 + \dots + \tfrac18\right)
   + \tfrac19\left(\tfrac19 + \dots + \tfrac1{15}\right) + \dots

</math>
or
<math>

\gamma = \sum_{k=2}^{\infty} \frac{k - \lfloor\sqrt{k}\rfloor^2}{k^2\lfloor\sqrt{k}\rfloor^2}
=  \tfrac1{2^2} + \tfrac2{3^2}
 + \tfrac1{2^2}\left(\tfrac1{5^2} + \tfrac2{6^2} + \tfrac3{7^2} + \tfrac4{8^2}\right)
 + \tfrac1{3^2}\left(\tfrac1{10^2} + \dots + \tfrac6{15^2}\right) + \dots

</math>
(see Krämer, 2005) Vacca's series may be obtained by manipulation of Catalan's 1875 integral (see Sondow and Zudilin)

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

The continued fraction expansion of <math> \gamma </math> is:

<math> \gamma = [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...]\, </math> (sequence A002852 in OEIS).

Asymptotic expansions

γ equals the following asymptotic formulas (where <math>H_n</math> is the nth harmonic number.)

<math>\gamma \sim H_n - \ln \left( n \right) - \frac{1} + \frac{1} - \frac{1} + ...</math>

(Euler)

<math>\gamma \sim H_n - \ln \left( {n + \frac{1}{2} + \frac{1} - \frac{1} + ...} \right)</math>

(Negoi)

<math>\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 }} - ...</math>

(Cesaro)

The third formula is also called the Ramanujan expansion.

e to the power γ

The constant e^γ is important in number theory. Some authors denote this quantity simply as <math> \gamma' </math>. e^γ equals the following limit, where p_n is the n-th prime number:

<math>

e^\gamma = \lim_{n \to \infty} \frac {1} {\log p_n} \prod_{i=1}^n \frac {p_i} {p_i - 1}. </math> This restates the third of Mertens' theorems. The numerical value of e^γ is:

<math>e^\gamma =1.78107241799019798523650410310717954916964521430343\dots</math>

Other infinite products relating to e^γ include:

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

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

These products result from the Barnes G-function. We also have

<math> 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 </math> where the nth factor is the (n+1)st root of

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

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

Generalizations

Euler's generalized constants are given by

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

for 0 < α < 1, with γ as the special case α = 1.^[3] This can be further generalized to

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

for some arbitrary decreasing function f. For example,

gives rise to the Stieltjes constants, and

gives

<math>\gamma_{f_a} = \frac{(a-1)\zeta(a)-1}{a-1}</math>

where again the limit

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

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

Known 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. (For the 20th digit, he calculated 1811209008239 when the correct value is 0651209008240.) In recent decades, faster computers and algorithms have dramatically increased the number of calculated digits in the decimal expansion of γ.^[4]

**Number of known decimal digits of γ**
Date	Decimal digits	Computation performed by
1734	5	Leonhard Euler
1736	15	Leonhard Euler
1790	19	Lorenzo Mascheroni
1809	24	Johann G. von Soldner
1812	40	F.B.G. Nicolai
1861	41	Oettinger
1869	59	William Shanks
1871	110	William Shanks
1878	263	John C. Adams
1962	1,271	Donald E. Knuth
1962	3,566	D.W. Sweeney
1977	20,700	Richard P. Brent
1980	30,100	Richard P. Brent & Edwin M. McMillan
1993	172,000	Jonathan Borwein
1997	1,000,000	Thomas Papanikolaou
December 1998	7,286,255	Xavier Gourdon
October 1999	108,000,000	Xavier Gourdon & Patrick Demichel
December 8, 2006	116,580,041	Alexander J. Yee^[5]
January 1, 2008	1,001,262,777	Richard B. Kreckel (claimed)^[6]
January 3, 2008	131,151,000	Nicholas D. Farrer (claimed)^[7]

Cultural appearances

The radio show Car Talk puzzler for the week of 23 October 2006 involved a car which repeatedly slows down as it nears its destination. The answer employed the Euler-Mascheroni constant to approximate the harmonic number <math>H_{75}</math>, the desired result.