|List of numbers – Irrational and suspected irrational numbers
|Continued fraction||[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, … ] (This continued fraction is not 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 (gamma).
It is defined as the limiting diffenence between the harmonic series and the natural logarithm:
Its numerical value to 50 decimal places is
0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 … (sequence A001620 in OEIS).
should not be confused with the base of the natural logarithm, , which is sometimes called Euler’s number.
The constant first appeared in a 1735 paper by the Swiss mathematician Leonard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascherori used the notations A and a for constant. The notation 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 in 1835.
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 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
- In Dimansional regularization of Feynman diagrams in Quantum Field Theory
- The infomation entropy of the Weibull and Lévy distributions
- The answer to the Coupon collector’s problem*
For more infomation of this nature, see Gourdon and Sebah (2004)
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 . The ubiquity of revealed by the large number of equations blow makes the irrationality of a major open question in mathematics. Also see Sondow
For more quations of the sortshown below, see Gourdon and Sebeh (2002).
Relation to gamma function
is related to digamma function , and hence the derivative of the gamma function , when both functions are evaluated at 1. Thus :
This is equal to limits :
Further limit results are (Krämer, 2005):
A limit related to the Beta function (expressed in terms of gamma functions) is
Relation to zeta function
can also be expressed as an infinite sum who se terms involve the Riemann zeta function evluated at positive intergers :
Other series related to the zeta function include :
The error term in the last quation is a rapidly decreasing function of n. As a result, the formula is well-suited for effcient computation of the constant to hight precison.
Other interesting limits equaling the Euler-Mascheroni constant are the antisymmetric limit (Sondow, 1998)
Closely related to this is the rational zeta series expression. By peeling off the first few terms of the series above, one obitans an estimate for the clasical series limit :
Where is the Hurwitz zeta function. The sum in this equation in volves the harmonic numbers, . Expanding some of the terms in the Hurwits zeta function gives :
equals the value of 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 serres (A094640)
It shows that may be thought of as an “alternating Euler constant”.
The two constants are also ralated by the pair of series (see Sondow 2005 #2)
Where and 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 fllowing infinite series appoaches :
The series for is equivalent to series nielsen found in 1897 :
In 1910, Vacca found the closely related series :
where is the logarithm to 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 :
equals the fllowing asymptotic formulas (where is the nth harmonic number.)
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 intergration. The coefficients are called Gregory coefficients; the first six were given in a latter to John Collins in 1670. From the recursin
we get the table (A002206 and A002207)
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 is important in number theory. Some authors denote this quantity simply as equals the following limit, where is the n-th prime number :
This restates the third of Mertens’ theorems. The numerical value of is A073004
Other infinite products relating to include :
These products result from the Barnes G-function.
We also have
where the n-th 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 (sequence A002852 in OEIS), and has at least 470,000 terms.
Euler’s generalized constants are given by for with as the special case This can be further generalized to
for some arbitrary decreasing function f. For example,
gives rise to the Stieltjes constants, and
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.)
Published Decimal Expansions of
|1811||22||Carl Friedrich Gauss|
|1811||22||Carl Friedrich Gauss|
|1812||40||Friedrich Bernhard Gottfried Nicolai|
|1857||34||Christian Fredrik Lindman|
|1871||99||James W.L. Glaisher|
|1878||263||John C. Adams|
|1952||329||John William Wrench, Jr.|
|1961||1050||Helmut Fischer and Carl Zeller|
|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|
|2009||29,844,489,545||Alexander J. Yee & Raymond Chan|