Euler–Mascheroni constant

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The area of the blue region converges to the 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)
Question, Web Fundamentals.svg Unsolved problem in mathematics:
Is Euler's constant irrational? If so, is it transcendental?
(more unsolved problems in mathematics)
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:

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. For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835 (, "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 (, "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. 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 ().

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 Murti 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 Murti and A. Zaytseva again considered an infinite list of numbers containing γ and showed than all but at most one are transcendental.[clarification needed]

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 ():

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 ():

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 () and () with equivalent series:

An interesting comparison by () 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 ()

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 )

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 (; , 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 (, ):

In 1910, Vacca found the closely related series (,[citation not found] , , ,[citation not found] , , )

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 () we get:

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

where Gn are Gregory coefficients (, , ) This series is the special case of the expansions

convergent for

A similar series with the Cauchy numbers of the second kind Cn is (; , 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 OEISA302120 and OEISA302121. Other series with the same polynomials include these examples:

and

where Γ(a) is the gamma function ().

A series related to the Akiyama-Tanigawa algorithm is

where Gn(2) are the Gregory coefficients of the second order ().

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.

, pp. 147–148 derived closed-form expressions for the sums of errors of these approximations. He showed that (Theorem A.1):

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 (). The numerical value of eγ is:

1.78107241799019798523650410310717954916964521430343... OEISA073004.

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 () using hypergeometric functions.

It also holds that

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, ...] OEISA002852, 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.

Generalizations[]

abm(x) = γx

Euler's generalized constants are given by

for 0 < α < 1, with γ as the special case α = 1 (, 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 .

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

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
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
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

Notes[]

  1. ^ Lagarias, Jeffrey C. (2013-07-19). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv:1303.1856. doi:10.1090/S0273-0979-2013-01423-X. ISSN 0273-0979. S2CID 119612431.
  2. ^ 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. 1423: 338–350. doi:10.1007/bfb0054873. ISBN 978-3-540-69113-6.
  3. Papanikolaou, T. (1997). Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie (Thesis). Universität des Saarlandes.
  4. Mahler, Kurt; Mordell, Louis Joel (1968-06-04). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173. Bibcode:1968RSPSA.305..149M. doi:10.1098/rspa.1968.0111. S2CID 123486171.
  5. Aptekarev, A. I. (2009-02-28). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].
  6. Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.
  7. Murty, M. Ram; Saradha, N. (2010-12-01). "Euler–Lehmer constants and a conjecture of Erdös". Journal of Number Theory. 130 (12): 2671–2682. doi:10.1016/j.jnt.2010.07.004. ISSN 0022-314X.
  8. Murty, M. Ram; Zaytseva, Anastasia (2013). "Transcendence of Generalized Euler Constants". The American Mathematical Monthly. 120 (1): 48–54. doi:10.4169/amer.math.monthly.120.01.048. ISSN 0002-9890.
  9. Choi, Junesang; Srivastava, H. M. (2010-09-01). "Integral Representations for the Euler–Mascheroni Constant γ". Integral Transforms and Special Functions. 21 (9): 675–690. doi:10.1080/10652461003593294. ISSN 1065-2469. S2CID 123698377.
  10. Yee, Alexander J. (March 7, 2011). "Large Computations". www.numberworld.org.
  11. ^ Yee, Alexander J. "Records Set by y-cruncher". www.numberworld.org. Retrieved April 30, 2018.
    Yee, Alexander J. "y-cruncher - A Multi-Threaded Pi-Program". www.numberworld.org.
  12. "Euler-Mascheroni Constant". Polymath Collector.

References[]

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