List of numbers – Irrational and suspected irrational numbers γ – ζ(3) – √2 – √3 – √5 – φ – ρ – δS – α – e – π – δ |
|
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, … ][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:
Here, ⌊x⌋ represents the floor function. The numerical value of this constant, to 50 decimal places, is
should not be confused with the base of the natural logarithm, e, which is sometimes called Euler's number.
Contents |
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]
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).
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).
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:
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
γ 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
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:
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:
γ equals the value of a number of definite integrals:
where is the fractional Harmonic number.
Definite integrals in which γ appears include:
One can express γ 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)
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 to the base 2 and is the floor function.
In 1926 he found a second series:
From the Kummer-expansion of the gamma function we get:
Series of prime numbers:
γ equals the following asymptotic formulas (where is the nth harmonic number.)
The third formula is also called the Ramanujan expansion.
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 recursion
we get the table
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | OEIS sequence |
---|---|---|---|---|---|---|---|---|---|---|---|
Cn | A002206 (numerators), A002207 (denominators) |
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:
The constant eγ is important in number theory. Some authors denote this quantity simply as . eγ equals the following limit, where pn is the nth prime number:
This restates the third of Mertens' theorems. The numerical value of eγ is:
Other infinite products relating to eγ 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.
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, ...] (sequence A002852 in OEIS), and has at least 470,000 terms.[4]
Euler's generalized constants are given by
for 0 < α < 1, with γ as the special case α = 1.[6] 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 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[7] |