Euler–Mascheroni constant

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 the floor function.

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

0.57721566490153286060651209008240243104215933593992...(sequence A001620 in the OEIS)
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) and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842 (De Morgan 1836–1842, "γ" on p. 578)

Appearances

The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):

Properties

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.[1][2] 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

where are ceiling brackets.

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 0 < ε < 1/252n6.

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 Hx 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 N1(n) and N0(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 MalmstenKummer 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 Gn are Gregory coefficients (Krämer 2005, Blagouchine 2016, Blagouchine 2018) This series is the special case of the expansions

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

see OEIS: A302120 and OEIS: A302121. Other series with the same polynomials include these examples:

and

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

A series related to the Akiyama-Tanigawa algorithm is

where Gn(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 nth 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 pn is the nth prime number:

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

1.78107241799019798523650410310717954916964521430343... OEIS: A073004.

Other infinite products relating to eγ include:

These products result from the Barnes G-function.

In addition,

where the nth 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 475,006 terms,[1] 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 ...1811209008239 when the correct value is ...0651209008240.

Published Decimal Expansions of γ
DateDecimal digitsAuthorSources
17345Leonhard Euler
173515Leonhard Euler
178116Leonhard Euler
179032Lorenzo Mascheroni, with 20-22 and 31-32 wrong
180922Johann G. von Soldner
181122Carl Friedrich Gauss
181240Friedrich Bernhard Gottfried Nicolai
185734Christian Fredrik Lindman
186141Ludwig Oettinger
186749William Shanks
187199James W.L. Glaisher
1871101William Shanks
1877262J. C. Adams
1952328John William Wrench Jr.
19611050Helmut Fischer and Karl Zeller
19621271Donald Knuth
19623566Dura W. Sweeney
19734879William A. Beyer and Michael S. Waterman
197720700Richard P. Brent
198030100Richard P. Brent & Edwin M. McMillan
1993172000Jonathan Borwein
1999108000000Patrick Demichel and Xavier Gourdon
200929844489545Alexander J. Yee & Raymond ChanYee 2011, y-cruncher 2017
2013119377958182Alexander J. YeeYee 2011, y-cruncher 2017
2016160000000000Peter Trueby-cruncher 2017
2016250000000000Ron Watkinsy-cruncher 2017
2017477511832674Ron Watkinsy-cruncher 2017

References

  1. Haible, Bruno; Papanikolaou, Thomas (1998). Buhler, Joe P. (ed.). "Fast multiprecision evaluation of series of rational numbers". Algorithmic Number Theory. Lecture Notes in Computer Science. Springer Berlin Heidelberg: 338–350. doi:10.1007/bfb0054873. ISBN 978-3-540-69113-6.

Further reading

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