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Euler's constant

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Euler's constant
γ
0.57721...[1]
General information
TypeUnknown
Fields
History
Discovered1734
ByLeonhard Euler
First mentionDe Progressionibus harmonicis observationes
Named after
The area of the blue region converges to Euler's constant.

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:

Here, ⌊·⌋ represents the floor function.

The numerical value of Euler's constant, to 50 decimal places, is:[1]

0.57721566490153286060651209008240243104215933593992...

History

[edit]

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43), where he described it as "worthy of serious consideration".[2][3] Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations C and O for the constant. The Italian mathematician Lorenzo 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. In 1790, he used the notations A and a for the constant. Other computations were done by Johann von Soldner in 1809, who used the notation H. 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.[3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835,[4] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[5] Euler's constant was also studied by the Indian mathematician Srinivasa Ramanujan who published one paper on it in 1917.[6] David Hilbert mentioned the irrationality of γ as an unsolved problem that seems "unapproachable" and, allegedly, the English mathematician Godfrey Hardy offered to give up his Savilian Chair at Oxford to anyone who could prove this.[2]

Appearances

[edit]

Euler's constant appears frequently in mathematics, especially in number theory and analysis.[7] Examples include, among others, the following places: (where '*' means that this entry contains an explicit equation):

Analysis

[edit]

Number theory

[edit]

In other fields

[edit]

Properties

[edit]

Irrationality and transcendence

[edit]

The number γ has not been proved algebraic or transcendental. In fact, it is not even known whether γ is irrational. The ubiquity of γ revealed by the large number of equations below and the fact that γ has been called the third most important mathematical constant after π and e[37][12] makes the irrationality of γ a major open question in mathematics.[2][38][39][32]

Unsolved problem in mathematics:
Is Euler's constant irrational? If so, is it transcendental?

However, some progress has been made. In 1959 Andrei Shidlovsky proved that at least one of Euler's constant γ and the Gompertz constant δ is irrational;[40][27] Tanguy Rivoal proved in 2012 that at least one of them is transcendental.[41] Kurt Mahler showed in 1968 that the number π/2 - γ is transcendental (with and being Bessel functions).[42][3] It is known that the transcendence degree of the field is at least two.[3] In 2010, M. Ram Murty and N. Saradha showed that at most one of the Euler-Lehmer constants, i. e. the numbers of the form

is algebraic, given that q ≥ 2 and 1 ≤ a < q; this family includes the special case γ(2,4) = γ/4.[3][43] Using the same approach, in 2013, M. Ram Murty and A. Zaytseva proved that a different family containing γ,[44]

which is based on sums of reciprocals of integers not divisible by a fixed list of primes Ω (given 1Ω = 1 if gcd(n, p) = 1 for all p ∈ Ω and 0 otherwise), has the same property.[3][45]

Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10244663.[46][47] If eγ is a rational number, then its denominator must be greater than 1015000.[3]

Euler's constant is conjectured not to be an algebraic period,[3] but the values of its first 109 decimal digits seem to indicate that it could be a normal number.[48]

Continued fraction

[edit]

The simple continued fraction expansion of Euler's constant is given by:[49]

which has no apparent pattern. It is known to have at least 16,695,000,000 terms,[49] and it has infinitely many terms if and only if γ is irrational.

The Khinchin limits for (red), (blue) and (green).

Numerical evidence suggests that both Euler's constant γ as well as the constant eγ are among the numbers for which the geometric mean of their simple continued fraction terms converges to Khinchin's constant. Similarly, when are the convergents of their respective continued fractions, the limit appears to converge to Lévy's constant in both cases.[50] However neither of these limits has been proven.[51]

There also exists a generalized continued fraction for Euler's constant.[52]

A good simple approximation of γ is given by the reciprocal of the square root of 3 or about 0.57735:[53]

with the difference being about 1 in 7,429.

Formulas and identities

[edit]

Relation to gamma function

[edit]

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:[54]

A limit related to the beta function (expressed in terms of gamma functions) is

Relation to the zeta function

[edit]

γ can also be expressed as an infinite sum whose terms involve the Riemann zeta function evaluated at positive integers:

The constant can also be expressed in terms of the sum of the reciprocals of non-trivial zeros of the zeta function:[55]

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 Euler's constant are the antisymmetric limit:[56]

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 k less than n, the average fraction by which the quotient n/k falls short of the next integer tends to γ (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 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:

Relation to triangular numbers

[edit]

Numerous formulations have been derived that express in terms of sums and logarithms of triangular numbers.[57][58][59][60] One of the earliest of these is a formula[61][62] for the th harmonic number attributed to Srinivasa Ramanujan where is related to in a series that considers the powers of (an earlier, less-generalizable proof[63][64] by Ernesto Cesàro gives the first two terms of the series, with an error term):

From Stirling's approximation[57][65] follows a similar series:

The series of inverse triangular numbers also features in the study of the Basel problem[66][67] posed by Pietro Mengoli. Mengoli proved that , a result Jacob Bernoulli later used to estimate the value of , placing it between and . This identity appears in a formula used by Bernhard Riemann to compute roots of the zeta function,[68] where is expressed in terms of the sum of roots plus the difference between Boya's expansion and the series of exact unit fractions :

Integrals

[edit]

γ equals the value of a number of definite integrals:

where Hx is the fractional harmonic number, and is the fractional part of .

The third formula in the integral list can be proved in the following way:

The integral on the second line of the equation stands for the Debye function value of +∞, which is m!ζ(m + 1).

Definite integrals in which γ appears include:[2][13]

We also have Catalan's 1875 integral[69]

One can express γ using a special case of Hadjicostas's formula as a double integral[39][70] with equivalent series:

An interesting comparison by Sondow[70] 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[71]

where N1(n) and N0(n) are the number of 1s and 0s, respectively, in the base 2 expansion of n.

Series expansions

[edit]

In general,

for any α > −n. However, the rate of convergence of this expansion depends significantly on α. In particular, γn(1/2) exhibits much more rapid convergence than the conventional expansion γn(0).[72][73] 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:[54][74]

In 1910, Vacca found the closely related series[75][76][77][78][79][54][80]

where log2 is the logarithm to base 2 and   is the floor function.

This can be generalized to:[81]

where:

In 1926 Vacca found a second series:

From the MalmstenKummer expansion for the logarithm of the gamma function[13] we get:

Ramanujan, in his lost notebook gave a series that approaches γ[82]:

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

where Gn are Gregory coefficients.[54][80][83] This series is the special case k = 1 of the expansions

convergent for k = 1, 2, ...

A similar series with the Cauchy numbers of the second kind Cn is[80][84]

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[85][86]

Other series with the same polynomials include these examples:

and

where Γ(a) is the gamma function.[83]

A series related to the Akiyama–Tanigawa algorithm is

where Gn(2) are the Gregory coefficients of the second order.[83]

As a series of prime numbers:

Asymptotic expansions

[edit]

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.

Alabdulmohsin derived closed-form expressions for the sums of errors of these approximations.[84] He showed that (Theorem A.1):

Exponential

[edit]

The constant eγ is important in number theory. Its numerical value is:[87]

1.78107241799019798523650410310717954916964521430343....

eγ equals the following limit, where pn is the nth prime number:

This restates the third of Mertens' theorems.[88]

We further have the following product involving the three constants e, π and γ:[29]

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.[89]

It also holds that[90]

Published digits

[edit]
Published decimal expansions of γ
Date Decimal digits Author Sources
1734 5 Leonhard Euler [3]
1735 15 Leonhard Euler [3]
1781 16 Leonhard Euler [3]
1790 32 Lorenzo Mascheroni, with 20–22 and 31–32 wrong [3]
1809 22 Johann G. von Soldner [3]
1811 22 Carl Friedrich Gauss [3]
1812 40 Friedrich Bernhard Gottfried Nicolai [3]
1861 41 Ludwig Oettinger [91]
1867 49 William Shanks [92]
1871 100 James W.L. Glaisher [3]
1877 263 J. C. Adams [3]
1952 328 John William Wrench Jr. [3]
1961 1050 Helmut Fischer and Karl Zeller [93]
1962 1271 Donald Knuth [94]
1963 3566 Dura W. Sweeney [95]
1973 4879 William A. Beyer and Michael S. Waterman [96]
1977 20700 Richard P. Brent [50]
1980 30100 Richard P. Brent & Edwin M. McMillan [97]
1993 172000 Jonathan Borwein [98]
1997 1000000 Thomas Papanikolaou [98]
1998 7286255 Xavier Gourdon [98]
1999 108000000 Patrick Demichel and Xavier Gourdon [98]
March 13, 2009 29844489545 Alexander J. Yee & Raymond Chan [99][100]
December 22, 2013 119377958182 Alexander J. Yee [100]
March 15, 2016 160000000000 Peter Trueb [100]
May 18, 2016 250000000000 Ron Watkins [100]
August 23, 2017 477511832674 Ron Watkins [100]
May 26, 2020 600000000100 Seungmin Kim & Ian Cutress [100][101]
May 13, 2023 700000000000 Jordan Ranous & Kevin O'Brien [100]
September 7, 2023 1337000000000 Andrew Sun [100]

Generalizations

[edit]

Stieltjes constants

[edit]
Euler's generalized constants abm(-) for α > 0.

Euler's generalized constants are given by

for 0 < α < 1, with γ as the special case α = 1.[102] Extending for α > 1 gives:

with again the limit:

This can be further generalized to

for some arbitrary decreasing function f. Setting

gives rise to the Stieltjes constants , that occur in the Laurent series expansion of the Riemann zeta function:

with

n approximate value of γn OEIS
0 +0.5772156649015 A001620
1 −0.0728158454836 A082633
2 −0.0096903631928 A086279
3 +0.0020538344203 A086280
4 +0.0023253700654 A086281
100 −4.2534015717080 × 1017
1000 −1.5709538442047 × 10486

Euler-Lehmer constants

[edit]

Euler–Lehmer constants are given by summation of inverses of numbers in a common modulo class:[43]

The basic properties are

and if the greatest common divisor gcd(a,q) = d then

Masser-Gramain constant

[edit]

A two-dimensional generalization of Euler's constant is the Masser-Gramain constant. It is defined as the following limiting difference:[103]

where is the smallest radius of a disk in the complex plane containing at least Gaussian integers.

The following bounds have been established: .[104]

See also

[edit]

References

[edit]
  • Bretschneider, Carl Anton (1837) [1835]. "Theoriae logarithmi integralis lineamenta nova". Crelle's Journal (in Latin). 17: 257–285.
  • Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. ISBN 978-0-691-09983-5.
  • Lagarias, Jeffrey C. (2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 556. arXiv:1303.1856. doi:10.1090/s0273-0979-2013-01423-x. S2CID 119612431.

Footnotes

[edit]
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  3. ^ a b c d e f g h i j k l m n o p q r Lagarias 2013.
  4. ^ Bretschneider 1837, "γ = c = 0,5772156649015328606181120900823..." on p. 260.
  5. ^ De Morgan, Augustus (1836–1842). The differential and integral calculus. London: Baldwin and Craddoc. "γ" on p. 578.
  6. ^ Brent, Richard P. (1994). "Ramanujan and Euler's Constant" (PDF). Proc. Symp. Applied Math. 48: 541–545.
  7. ^ Sondow, Jonathan (2004). "The Euler constant: γ". Retrieved 2024-11-01.
  8. ^ Davis, P. J. (1959). "Leonhard Euler's Integral: A Historical Profile of the Gamma Function". American Mathematical Monthly. 66 (10): 849–869. doi:10.2307/2309786. JSTOR 2309786. Archived from the original on 7 November 2012. Retrieved 3 December 2016.
  9. ^ "DLMF: §5.17 Barnes' 𝐺-Function (Double Gamma Function) ‣ Properties ‣ Chapter 5 Gamma Function". dlmf.nist.gov. Retrieved 2024-11-01.
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  13. ^ a b c Blagouchine, Iaroslav V. (2014-10-01). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. ISSN 1572-9303.
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Further reading

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