Euler–Mascheroni constant
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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 ().
It is defined as the limiting difference between the harmonic series and the natural logarithm:
Here, 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) |
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 appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function.[3] For example, the German mathematician Carl Anton Bretschneider used the notation
in 1835[4] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[5]
Appearances
The Euler–Mascheroni constant appears, among other places, in the following ('*' 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
- In Dimensional regularization of Feynman diagrams in Quantum Field Theory
- The calculation of the Meissel–Mertens constant
- The third of Mertens' theorems*
- Solution of the second kind to Bessel's equation
- In the regularization/renormalization of the Harmonic series as a finite value
- The mean of the Gumbel distribution
- The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
- The answer to the coupon collector's problem*
- In some formulations of Zipf's law
- A definition of the cosine integral*
- Lower bounds to a prime gap.
Properties
The number has not been proved algebraic or transcendental. In fact, it is not even known whether
is irrational. Continued fraction analysis reveals that if
is rational, its denominator must be greater than 10242080.[6] The ubiquity of
revealed by the large number of equations below makes the irrationality of
a major open question in mathematics. Also see Sondow (2003a).
Relation to gamma function
is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:
This is equal to the limits:
Further limit results are (Krämer, 2005):
A limit related to the beta function (expressed in terms of gamma functions) is
Relation to the zeta function
can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:
Other series related to the zeta function include:
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)
and de la Vallée-Poussin's formula
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:
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:
, where
Integrals
equals the value of a number of definite integrals:
where is the fractional Harmonic number.
Definite integrals in which appears include:
One can express using a special case of Hadjicostas's formula as a double integral (Sondow 2003a, 2005) with equivalent series:
An interesting comparison by J. Sondow (2005) is the double integral and alternating series
It shows that may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series (see Sondow 2005 #2)
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)
Series expansions
Euler showed that the following infinite series approaches :
The series for is equivalent to series Nielsen found in 1897:
In 1910, Vacca found the closely related series:
where is the logarithm of base 2 and
is the floor function.
In 1926 he found a second series:
From the Malmsten-Kummer-expansion for the logarithm of the gamma function we get:
Series of prime numbers:
Series relating to square roots:
Asymptotic expansions
equals the following asymptotic formulas (where
is the nth harmonic number.)
- (Euler)
- (Negoi)
- (Cesàro)
The third formula is also called the Ramanujan expansion.
Relations with the reciprocal logarithm
The reciprocal logarithm function (Krämer, 2005)
has a deep connection with Euler's constant and was studied by James Gregory in connection with numerical integration. The coefficients are called Gregory coefficients; the first six were given in a letter to John Collins in 1670. From the equations
which can be used recursively to get these coefficients for all , we get the table
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | OEIS sequences |
---|---|---|---|---|---|---|---|---|---|---|---|
Cn | ![]() |
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Gregory coefficients are similar to Bernoulli numbers and satisfy the asymptotic relation
and the integral representation
Euler's constant has the integral representations
A very important expansion of Gregorio Fontana (1780) is:
which is convergent for all n.
Weighted sums of the Gregory coefficients give different constants:
Exponential
The constant is important in number theory. Some authors denote this quantity simply as
.
equals the following limit, where pn is the nth prime number:
This restates the third of Mertens' theorems.[8] The numerical value of is:
- 1.78107241799019798523650410310717954916964521430343 …
A073004.
Other infinite products relating to include:
These products result from the Barnes G-function.
We also have
where the nth factor is the (n+1)st root of
This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.
Continued fraction
The continued fraction expansion of 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
is irrational.
Generalizations
Euler's generalized constants are given by
for 0 < α < 1, with as the special case α = 1.[9] This can be further generalized to
for some arbitrary decreasing function f. For example,
gives rise to the Stieltjes constants, and
gives
where again the limit
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]
The basic properties are
and if gcd(a,q)=d then
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.
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
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Notes
- Footnotes
- ↑
A001620
- ↑
A002852
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 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])
- ↑ Augustus De Morgan: The differential and integral calculus, Baldwin and Craddock, London 1836–1842 ("γ" on p. 578)
- ↑ 6.0 6.1 Havil 2003 p 97.
- ↑ http://mathworld.wolfram.com/Euler-MascheroniConstant.html
- ↑ http://mathworld.wolfram.com/MertensConstant.html (14)
- ↑ Havil, 117-118
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 12.0 12.1 Nagisa – Large Computations
- References
- Lua error in package.lua at line 80: module 'strict' not found. PDF
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found. 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.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found. with an Appendix by Sergey Zlobin
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found. Ramanujan Journal 12: 225-244.
- Lua error in package.lua at line 80: module 'strict' not found.
- James Whitbread Lee Glaisher (1872), "On the history of Euler's constant". Messenger of Mathematics vol.1, p. 25-30, JFM 03.0130.01
- Lua error in package.lua at line 80: module 'strict' not found. (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
- Lua error in package.lua at line 80: module 'strict' not found.
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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).