Here, represents the floor function.
The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:
Is Euler's constant irrational? If so, is it transcendental?
|Continued fraction||[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, ...]|
(It is not known whether this continued fraction is finite, infinite periodic or infinite non-periodic. Shown in linear notation)
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 perhaps because of the constant's connection to the gamma function. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.
The Euler–Mascheroni constant appears, among other places, in the following (where '*' 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 Laurent 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
- The asymptotic expansion of the gamma function for small arguments.
- 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
- An upper bound on Shannon entropy in quantum information theory
The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10244663. The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics.
However, some progress was made. Kurt Mahler showed in 1968 that the number is transcendental (here, and are Bessel functions). In 2009 Alexander Aptekarev proved that at least one of the Euler–Mascheroni constant γ and the Euler–Gompertz constant δ is irrational. This result was improved in 2012 by Tanguy Rivoal, who proved that at least one of them is transcendental.
In 2010 M. Ram Murty and N. Saradha considered an infinite list of numbers containing γ/4 and showed that all but at most one of them are transcendental. In 2013 M. Ram Murty and A. Zaytseva again considered an infinite list of numbers containing γ and showed that all but at most one are transcendental.[clarification needed]
Relation to gamma function
This is equal to the limits:
Further limit results are:
Relation to the zeta function
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:
and the following formula, established in 1898 by de la Vallée-Poussin:
where are ceiling brackets. This formula indicates that when taking any positive integer n and dividing it by each positive integer m less than n, the average fraction by which the quotient n/m falls short of the next integer tends to (rather than 0.5) as n tends to infinity.
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 0 < ε < 1/252n6.
γ can also be expressed as follows where A is the Glaisher–Kinkelin constant:
γ equals the value of a number of definite integrals:
where Hx is the fractional harmonic number.
Definite integrals in which γ appears include:
An interesting comparison by Sondow is the double integral and alternating series
It shows that log 4/π may be thought of as an "alternating Euler constant".
The two constants are also related by the pair of series
where N1(n) and N0(n) are the number of 1s and 0s, respectively, in the base 2 expansion of n.
Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.
Euler showed that the following infinite series approaches γ:
In 1926 he found a second series:
Blagouchine (2018) found an interesting generalisation of the Fontana-Mascheroni series
where ψn(a) are the Bernoulli polynomials of the second kind, which are defined by the generating function
Other series with the same polynomials include these examples:
A series related to the Akiyama-Tanigawa algorithm is
Series of prime numbers:
γ equals the following asymptotic formulas (where Hn is the nth harmonic number):
The third formula is also called the Ramanujan expansion.
Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations. He showed that (Theorem A.1):
Other infinite products relating to eγ include:
These products result from the Barnes G-function.
where the nth factor is the (n + 1)th root of
It also holds that
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, ...], which has no apparent pattern. The continued fraction is known to have at least 475,006 terms, and it has infinitely many terms if and only if γ is irrational.
Euler's generalized constants are given by
for 0 < α < 1, with γ as the special case α = 1. This can be further generalized to
for some arbitrary decreasing function f. For example,
gives rise to the Stieltjes constants, and
where again the limit
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:
The basic properties are
and if gcd(a,q) = d then
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 and 31st-32nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.
|1790||32||Lorenzo Mascheroni, with 20-22 and 31-32 wrong|
|1809||22||Johann G. von Soldner|
|1811||22||Carl Friedrich Gauss|
|1812||40||Friedrich Bernhard Gottfried Nicolai|
|1857||34||Christian Fredrik Lindman|
|1871||99||James W.L. Glaisher|
|1877||262||J. C. Adams|
|1952||328||John William Wrench Jr.|
|1961||1050||Helmut Fischer and Karl Zeller|
|1962||3566||Dura W. Sweeney|
|1973||4879||William A. Beyer and Michael S. Waterman|
|1977||20700||Richard P. Brent|
|1980||30100||Richard P. Brent & Edwin M. McMillan|
|1999||108000000||Patrick Demichel and Xavier Gourdon|
|March 13, 2009||29844489545||Alexander J. Yee & Raymond Chan|||
|December 22, 2013||119377958182||Alexander J. Yee|||
|March 15, 2016||160000000000||Peter Trueb|||
|May 18, 2016||250000000000||Ron Watkins|||
|August 23, 2017||477511832674||Ron Watkins|||
|May 26, 2020||600000000100||Seungmin Kim & Ian Cutress|||
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