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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). Euler used the notations C and O for the constant. In 1790, the 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.[2] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835,[3] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[4]

Appearances

[edit]

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

Properties

[edit]

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.[7][8] The ubiquity of γ revealed by the large number of equations below makes the irrationality of γ a major open question in mathematics.[9]

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

However, some progress has been made. Kurt Mahler showed in 1968 that the number is transcendental (here, and are Bessel functions).[10][2] In 2009 Alexander Aptekarev proved that at least one of Euler's constant γ and the Euler–Gompertz constant δ is irrational;[11] Tanguy Rivoal proved in 2012 that at least one of them is transcendental.[12] It is known that the transcendence degree of the field is at least two.[2] In 2010 M. Ram Murty and N. Saradha showed that at most one of the numbers of the form

with q ≥ 2 and 1 ≤ a < q is algebraic; this family includes the special case γ(2,4) = γ/4.[2][13] In 2013 M. Ram Murty and A. Zaytseva found a different family containing γ, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.[2][14]

Euler's constant is conjectured not to be an algebraic period.[2]

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:[15]

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:[16]

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:[17]

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.[18][19][20][21] One of the earliest of these is a formula[22][23] 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[24][25] by Ernesto Cesàro gives the first two terms of the series, with an error term):

From Stirling's approximation[18][26] follows a similar series:

The series of inverse triangular numbers also features in the study of the Basel problem[27][28] 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,[29] 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:[30][31]

One can express γ using a special case of Hadjicostas's formula as a double integral[9][32] with equivalent series:

An interesting comparison by Sondow[32] 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[33]

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

We also have Catalan's 1875 integral[34]

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).[35][36] 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:[15][37]

In 1910, Vacca found the closely related series[38][39][40][41][42][15][43]

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

This can be generalized to:[44]

where:

In 1926 Vacca found a second series:

From the MalmstenKummer expansion for the logarithm of the gamma function[31] we get:

Ramanujan, in his lost notebook gave a series that approaches γ[45]:

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

where Gn are Gregory coefficients.[15][43][46] 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[43][47]

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[48][49]

Other series with the same polynomials include these examples:

and

where Γ(a) is the gamma function.[46]

A series related to the Akiyama–Tanigawa algorithm is

where Gn(2) are the Gregory coefficients of the second order.[46]

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.[47] He showed that (Theorem A.1):

Exponential

[edit]

The constant eγ is important in number theory. Its numerical value is:[50]

1.78107241799019798523650410310717954916964521430343....

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

This restates the third of Mertens' theorems.[51]

We further have the following product involving the three constants e, π and γ:[52]

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.[53]

It also holds that[54]

If eγ is a rational number, then its denominator must be greater than 1015000.[2]

Continued fraction

[edit]

The simple continued fraction expansion of Euler's constant is given by:[55]

which has no apparent pattern. It is known to have at least 4,800,000,000 terms,[30] and it has infinitely many terms if and only if γ is irrational.

Numerical evidence suggests that Euler's constant is among the numbers for which the geometric mean of the continued fraction terms converges to Khinchin's constant. Similarly, when are the convergents of the continued fraction, the limit appears to converge to Lévy's constant. However neither of these limits has been proven.[30]

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.[56] 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:[13]

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:[57]

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

The following bounds have been established: .[58]

Published digits

[edit]

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 [59]
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 [60][61]
December 22, 2013 119377958182 Alexander J. Yee [61]
March 15, 2016 160000000000 Peter Trueb [61]
May 18, 2016 250000000000 Ron Watkins [61]
August 23, 2017 477511832674 Ron Watkins [61]
May 26, 2020 600000000100 Seungmin Kim & Ian Cutress [61][62]
May 13, 2023 700000000000 Jordan Ranous & Kevin O'Brien [61]
September 7, 2023 1337000000000 Andrew Sun [61]

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]
  1. ^ a b Sloane, N. J. A. (ed.). "Sequence A001620 (Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ a b c d e f g Lagarias 2013.
  3. ^ Bretschneider 1837, "γ = c = 0,5772156649015328606181120900823..." on p. 260.
  4. ^ De Morgan, Augustus (1836–1842). The differential and integral calculus. London: Baldwin and Craddoc. "γ" on p. 578.
  5. ^ Caves, Carlton M.; Fuchs, Christopher A. (1996). "Quantum information: How much information in a state vector?". The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later. Israel Physical Society. arXiv:quant-ph/9601025. Bibcode:1996quant.ph..1025C. ISBN 9780750303941. OCLC 36922834.
  6. ^ Connallon, Tim; Hodgins, Kathryn A. (October 2021). "Allen Orr and the genetics of adaptation". Evolution. 75 (11): 2624–2640. doi:10.1111/evo.14372. PMID 34606622. S2CID 238357410.
  7. ^ Haible, Bruno; Papanikolaou, Thomas (1998). "Fast multiprecision evaluation of series of rational numbers". In Buhler, Joe P. (ed.). Algorithmic Number Theory. Lecture Notes in Computer Science. Vol. 1423. Springer. pp. 338–350. doi:10.1007/bfb0054873. ISBN 9783540691136.
  8. ^ Papanikolaou, T. (1997). Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie (Thesis) (in German). Universität des Saarlandes.
  9. ^ a b See also Sondow, Jonathan (2003). "Criteria for irrationality of Euler's constant". Proceedings of the American Mathematical Society. 131 (11): 3335–3344. arXiv:math.NT/0209070. doi:10.1090/S0002-9939-03-07081-3. S2CID 91176597.
  10. ^ Mahler, Kurt; Mordell, Louis Joel (4 June 1968). "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.
  11. ^ Aptekarev, A. I. (28 February 2009). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].
  12. ^ 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.
  13. ^ a b Ram Murty, M.; Saradha, N. (2010). "Euler–Lehmer constants and a conjecture of Erdos". Journal of Number Theory. 130 (12): 2671–2681. doi:10.1016/j.jnt.2010.07.004. ISSN 0022-314X.
  14. ^ 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. JSTOR 10.4169/amer.math.monthly.120.01.048. S2CID 20495981.
  15. ^ a b c d Krämer, Stefan (2005). Die Eulersche Konstante γ und verwandte Zahlen (in German). University of Göttingen.
  16. ^ Wolf, Marek (2019). "6+infinity new expressions for the Euler-Mascheroni constant". arXiv:1904.09855 [math.NT]. The above sum is real and convergent when zeros and complex conjugate are paired together and summed according to increasing absolute values of the imaginary parts of . See formula 11 on page 3. Note the typographical error in the numerator of Wolf's sum over zeros, which should be 2 rather than 1.
  17. ^ Sondow, Jonathan (1998). "An antisymmetric formula for Euler's constant". Mathematics Magazine. 71 (3): 219–220. doi:10.1080/0025570X.1998.11996638. Archived from the original on 2011-06-04. Retrieved 2006-05-29.
  18. ^ a b Boya, L.J. (2008). "Another relation between π, e, γ and ζ(n)". Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. 102 (2): 199–202. doi:10.1007/BF03191819. γ/2 in (10) reflects the residual (finite part) of ζ(1)/2, of course. See formulas 1 and 10.
  19. ^ Sondow, Jonathan (2005). "Double Integrals for Euler's Constant and and an Analog of Hadjicostas's Formula". The American Mathematical Monthly. 112 (1): 61–65. doi:10.2307/30037385. JSTOR 30037385. Retrieved 2024-04-27.
  20. ^ Chen, Chao-Ping (2018). "Ramanujan's formula for the harmonic number". Applied Mathematics and Computation. 317: 121–128. doi:10.1016/j.amc.2017.08.053. ISSN 0096-3003. Retrieved 2024-04-27.
  21. ^ Lodge, A. (1904). "An approximate expression for the value of 1 + 1/2 + 1/3 + ... + 1/r". Messenger of Mathematics. 30: 103–107.
  22. ^ Villarino, Mark B. (2007). "Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number". arXiv:0707.3950 [math.CA]. It would also be interesting to develop an expansion for n! into powers of m, a new Stirling expansion, as it were. See formula 1.8 on page 3.
  23. ^ Mortici, Cristinel (2010). "On the Stirling expansion into negative powers of a triangular number". Math. Commun. 15: 359–364.
  24. ^ Cesàro, E. (1885). "Sur la série harmonique". Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale (in French). 4. Carilian-Goeury et Vor Dalmont: 295–296.
  25. ^ Bromwich, Thomas John I'Anson (2005) [1908]. An Introduction to the Theory of Infinite Series (PDF) (3rd ed.). United Kingdom: American Mathematical Society. p. 460. See exercise 18.
  26. ^ Whittaker, E.; Watson, G. (2021) [1902]. A Course of Modern Analysis (5th ed.). p. 271, 275. doi:10.1017/9781009004091. ISBN 9781316518939. See Examples 12.21 and 12.50 for exercises on the derivation of the integral form of the series .
  27. ^ Lagarias 2013, p. 13.
  28. ^ Nelsen, R. B. (1991). "Proof without Words: Sum of Reciprocals of Triangular Numbers". Mathematics Magazine. 64 (3): 167. doi:10.1080/0025570X.1991.11977600.
  29. ^ Edwards, H. M. (1974). Riemann's Zeta Function. Pure and Applied Mathematics, Vol. 58. Academic Press. pp. 67, 159.
  30. ^ a b c Weisstein, Eric W. "Euler-Mascheroni Constant Continued Fraction". mathworld.wolfram.com. Retrieved 2024-09-23.
  31. ^ a b 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.
  32. ^ a b Sondow, Jonathan (2005). "Double integrals for Euler's constant and and an analog of Hadjicostas's formula". American Mathematical Monthly. 112 (1): 61–65. arXiv:math.CA/0211148. doi:10.2307/30037385. JSTOR 30037385.
  33. ^ Sondow, Jonathan (1 August 2005a). New Vacca-type rational series for Euler's constant and its 'alternating' analog . arXiv:math.NT/0508042.
  34. ^ Sondow, Jonathan; Zudilin, Wadim (2006). "Euler's constant, q-logarithms, and formulas of Ramanujan and Gosper". The Ramanujan Journal. 12 (2): 225–244. arXiv:math.NT/0304021. doi:10.1007/s11139-006-0075-1. S2CID 1368088.
  35. ^ DeTemple, Duane W. (May 1993). "A Quicker Convergence to Euler's Constant". The American Mathematical Monthly. 100 (5): 468–470. doi:10.2307/2324300. ISSN 0002-9890. JSTOR 2324300.
  36. ^ Havil 2003, pp. 75–8.
  37. ^ Blagouchine 2016.
  38. ^ Vacca, G. (1910). "A new analytical expression for the number π and some historical considerations". Bulletin of the American Mathematical Society. 16: 368–369. doi:10.1090/S0002-9904-1910-01919-4.
  39. ^ Glaisher, James Whitbread Lee (1910). "On Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 41: 365–368.
  40. ^ Hardy, G.H. (1912). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
  41. ^ Vacca, G. (1926). "Nuova serie per la costante di Eulero, C = 0,577...". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche". Matematiche e Naturali (in Italian). 6 (3): 19–20.
  42. ^ Kluyver, J.C. (1927). "On certain series of Mr. Hardy". Q. J. Pure Appl. Math. 50: 185–192.
  43. ^ a b c Blagouchine, Iaroslav V. (2016). "Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only". J. Number Theory. 158: 365–396. arXiv:1501.00740. doi:10.1016/j.jnt.2015.06.012.
  44. ^ Pilehrood, Khodabakhsh Hessami; Pilehrood, Tatiana Hessami (2008-08-04), Vacca-type series for values of the generalized-Euler-constant function and its derivative, doi:10.48550/arXiv.0808.0410, retrieved 2024-10-08
  45. ^ Berndt, Bruce C. (January 2008). "A fragment on Euler's constant in Ramanujan's lost notebook". South East Asian J. Math. & Math. Sc. 6 (2): 17–22.
  46. ^ a b c Blagouchine, Iaroslav V. (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions". INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A (#A3): 1–45. arXiv:1606.02044. Bibcode:2016arXiv160602044B.
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  50. ^ Sloane, N. J. A. (ed.). "Sequence A073004 (Decimal expansion of exp(gamma))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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Further reading

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