# Euler–Mascheroni constant

# Euler–Mascheroni constant

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, ...] (It is not known whether this continued fraction is finite, infinite periodic or infinite non-periodic. Shown in linear notation) Source: Sloane |

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 (γ).

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

`Here,represents thefloor function.`

The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:

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, ...] (It is not known whether this continued fraction is finite, infinite periodic or infinite non-periodic. Shown in linear notation) Source: Sloane |

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 perhaps because of the constant's connection to the gamma function (Lagarias 2013). For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 (Bretschneider 1837, "γ = *c* = 0,577215 664901 532860 618112 090082 3.." on p. 260 ^{[51]} ) and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842 (De Morgan 1836–1842, "γ" on p. 578 ^{[52]} )

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 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

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 (Caves & Fuchs 1996)

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 (Havil 2003, p. 97). 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).

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:

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

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 de la Vallée-Poussin's formula

`whereareceilingbrackets.`

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, *H**n*. Expanding some of the terms in the Hurwitz zeta function gives:

where 0 < *ε* < 1/252*n*6.

γ can also be expressed as follows where *A* is the Glaisher–Kinkelin constant:

γ can also be expressed as follows, which can be proven by expressing the zeta function as a Laurent series:

Integrals

γ equals the value of a number of definite integrals:

where *H**x* 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) and (Sondow 2005) with equivalent series:

An interesting comparison by (Sondow 2005) is the double integral and alternating series

It shows that ln 4/π may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series (Sondow 2005a)

where *N*1(*n*) and *N*0(*n*) are the number of 1s and 0s, respectively, in the base 2 expansion of *n*.

We have also Catalan's 1875 integral (see Sondow & Zudilin 2006)

Series expansions

In general,

`for any. However, the rate of convergence of this expansion depends significantly on. In particular,exhibits much more rapid convergence than the conventional expansion(DeTemple 1993;Havil 2003, pp. 75-78). This is because`

while

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:

The series for γ is equivalent to a series Nielsen found in 1897 (Krämer 2005, Blagouchine 2016):

In 1910, Vacca found the closely related series (Vacca 1910, Glaisher 1910, Hardy 1912, Vacca 1925, Kluyver 1927, Krämer 2005, Blagouchine 2016)

where log2 is the logarithm to 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 (Blagouchine 2014) we get:

An important expansion for Euler's constant is due to Fontana and Mascheroni

`where*G`

_{n}`convergent for`

A similar series with the Cauchy numbers of the second kind *Cn* is (Blagouchine 2016; Alabdulmohsin 2018, pp. 147-148)

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

For any rational *a* this series contains rational terms only. For example, at *a* = 1, it becomes

and

where Γ(*a*) is the gamma function (Blagouchine 2018).

A series related to the Akiyama-Tanigawa algorithm is

where *G**n*(2) are the Gregory coefficients of the second order (Blagouchine 2018).

Series of prime numbers:

Asymptotic expansions

γ equals the following asymptotic formulas (where *Hn* is the *n*th harmonic number):

- (

*Euler*)(

*Negoi*)(*Cesàro*)

The third formula is also called the Ramanujan expansion.

Exponential

The constant *eγ* is important in number theory. Some authors denote this quantity simply as *γ′*. *eγ* equals the following limit, where *p**n* is the *n*th prime number:

This restates the third of Mertens' theorems (Weisstein n.d.). The numerical value of *eγ* is:

Other infinite products relating to *eγ* include:

These products result from the Barnes *G*-function.

In addition,

where the *n*th factor is the (*n* + 1)th root of

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (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, ...] OEIS: A002852, which has no *apparent* pattern. The continued fraction is known to have at least 470,000 terms (Havil 2003, p. 97), and it has infinitely many terms if and only if γ is irrational.

Generalizations

*Euler's generalized constants* are given by

for 0 < *α* < 1, with γ as the special case *α* = 1 (Havil 2003, pp. 117–118). 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 (Ram Murty & Saradha 2010):

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 and 31st-32nd decimal places; starting from the 20th digit, he calculated ...**181**12090082**39** when the correct value is ...**065**12090082**40**.

Date | Decimal digits | Author | Sources |
---|---|---|---|

1734 | 5 | Leonhard Euler | |

1735 | 15 | Leonhard Euler | |

1781 | 16 | Leonhard Euler | |

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 | |

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 | 1271 | Donald Knuth | |

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 | |

1993 | 172000 | Jonathan Borwein | |

1999 | 108000000 | Patrick Demichel and Xavier Gourdon | |

2009 | 29844489545 | Alexander J. Yee & Raymond Chan | Yee 2011, y-cruncher 2017 |

2013 | 119377958182 | Alexander J. Yee | Yee 2011, y-cruncher 2017 |

2016 | 160000000000 | Peter Trueb | y-cruncher 2017 |

2016 | 250000000000 | Ron Watkins | y-cruncher 2017 |

2017 | 477511832674 | Ron Watkins | y-cruncher 2017 |