Euler–Mascheroni constant


Euler–Mascheroni constant
The area of the blue region is equal to the Euler–Mascheroni constant.
List of numbersIrrational and suspected irrational numbers
γζ(3)√2√3√5φρδSαeπδ
Binary 0.100100111100010001...
Decimal 0.5772156649015328606065...
Hexadecimal 0.93C467E37DB0C7A4D1BE...
Continued fraction [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, … ][1](This continued fraction is not known to be periodic. Shown in linear notation)

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 γ (gamma).

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

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

Here, ⌊x⌋ represents the floor function. The numerical value of this constant, to 50 decimal places, is

0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 … (sequence A001620 in OEIS).

γ should not be confused with the base of the natural logarithm, e, which is sometimes called Euler's number.

Contents

History

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and 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. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835[2] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[3]

Appearances

The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit 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 reveals that if γ is rational, its denominator must be greater than 10242080.[4] 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

γ 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

γ 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 )

and

\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 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} - \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

γ equals the value of a number of definite integrals:

\begin{align}\gamma &= - \int_0^\infty { e^{-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 Hx is the fractional Harmonic number.

Definite integrals in which γ appears include:

  \int_0^\infty { e^{-x^2} \ln x }\,dx = -\tfrac14(\gamma+2 \ln 2) \sqrt{\pi}
OEISA081855

One can express γ 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 (OEISA094640)

 \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 γ:

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

The series for γ 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 log2 is the logarithm to the 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 Kummer-expansion 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}. [5]

Asymptotic expansions

γ equals the following asymptotic formulas (where Hn 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.

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 Cn are called Gregory coefficients; the first six were given in a letter to John Collins in 1670. From the recursion

C_0 = -1,\quad \sum_{k=0}^{n-1}\frac{C_k}{n-k} = 0,\quad n=2,3,4,\dots,

we get the table (OEISA002206 and OEISA002207)

n 1 2 3 4 5 6 7 8 9 10
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}

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=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 n-th 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. The numerical value of eγ is OEISA073004:

e^\gamma =1.78107241799019798523650410310717954916964521430343\dots

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 γ is of the form [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (sequence A002852 in OEIS), and has at least 470,000 terms.[4]

Generalizations

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.[6] 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

fa(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.

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 γ
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[7]

Notes

  1. ^ Sloane's A002852 : Continued fraction for Euler's constant. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ Car. Ant. 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 p. 260)
  3. ^ Augustus De Morgan: The differential and integral calculus, Baldwin and Craddock, London 1836–1842 (“γ” on p. 578)
  4. ^ a b Havil 2003 p 97.
  5. ^ http://mathworld.wolfram.com/MertensConstant.html (15)
  6. ^ Havil, 117-118
  7. ^ Nagisa – Large Computations

References

External links


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