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词条 Euler–Mascheroni constant
释义

  1. History

  2. Appearances

  3. Properties

     Relation to gamma function  Relation to the zeta function  Integrals  Series expansions  Asymptotic expansions  Exponential  Continued fraction 

  4. Generalizations

  5. Published digits

  6. Notes

  7. External links

{{redirect|Euler's constant|the base of the natural logarithm, {{mvar|e}} ≈ 2.718...|e (mathematical constant){{!}}{{mvar|e}} (mathematical 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 ({{mvar|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:

{{gaps|0.57721|56649|01532|86060|65120|90082|40243|10421|59335|93992|...}} {{nowrap|{{OEIS|A001620}}}}

Binary0.1001|0011|1100|0100|0110|0111|1110|0011|0111|1101|...}}
Decimal0.57721|56649|01532|86060|65120|90082|40243|10421|...}}
Hexadecimal0.93C4|67E3|7DB0|C7A4|D1BE|3F81|0152|CB56|A1CE|CC3A|...}}
Continued fraction[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, ...]}}[1]
(It is not known whether this continued fraction is finite, infinite periodic or infinite non-periodic.
Shown in linear notation)

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 {{math|C}} and {{math|O}} for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations {{math|A}} and {{math|a}} for the constant. The notation {{mvar|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 {{mvar|γ}} in 1835[3] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[4]

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[5]

Properties

The number {{mvar|γ}} has not been proved algebraic or transcendental. In fact, it is not even known whether {{mvar|γ}} is irrational. Continued fraction analysis reveals that if {{mvar|γ}} is rational, its denominator must be greater than 10242080.[6] The ubiquity of {{mvar|γ}} revealed by the large number of equations below makes the irrationality of {{mvar|γ}} a major open question in mathematics. Also see Sondow (2003a).

Relation to gamma function

{{mvar|γ}} is related to the digamma function {{math|Ψ}}, and hence the derivative of the gamma function {{math|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

{{mvar|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 {{math|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 {{math|ζ(s,k)}} is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, {{math|Hn}}. Expanding some of the terms in the Hurwitz zeta function gives:

where {{math|0 < ε < {{sfrac|1|252n6}}.}}

{{mvar|γ}} can also be expressed as follows where {{math|A}} is the Glaisher–Kinkelin constant:

Integrals

{{mvar|γ}} equals the value of a number of definite integrals:

where {{math|Hx}} is the fractional harmonic number.

Definite integrals in which {{mvar|γ}} appears include:

One can express {{mvar|γ}} using a special case of Hadjicostas's formula as a double integral (Sondow 2003a, 2005) with equivalent series:

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

It shows that {{math|ln {{sfrac|4|π}}}} may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series (see Sondow 2005 #2)

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

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

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 .[6][7] 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 {{mvar|γ}}:

The series for {{mvar|γ}} is equivalent to a series Nielsen found in 1897:[8]

In 1910, Vacca found the closely related series[9]

where {{math|log2}} is the logarithm to base 2 and {{math|⌊ ⌋}} is the floor function.

In 1926 he found a second series:

From the Malmsten–Kummer expansion for the logarithm of the gamma function[10] we get:

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

where {{math|Gn}} are Gregory coefficients.[11] This series is the special case of the expansions

convergent for

A similar series with the Cauchy numbers of the second kind {{math|Cn}} is[12]

Blagouchine (2018) found an interesting generalisation of the Fontana-Mascheroni series

where {{math|ψn(a)}} are the Bernoulli polynomials of the second kind, which are defined by the generating function

For any rational {{math|a}} this series contains rational terms only. For example, at {{math|a {{=}} 1}}, it becomes

see OEIS {{OEIS2C|id=A302120}} and {{OEIS2C|id=A302121}}. Other series with the same polynomials include these examples:

and

where {{math|Γ(a)}} is the gamma function.[13]

A series related to the Akiyama-Tanigawa algorithm is

where {{math|Gn(2)}} are the Gregory coefficients of the second order.[13]

Series of prime numbers:

Asymptotic expansions

{{mvar|γ}} equals the following asymptotic formulas (where {{math|Hn}} is the {{math|n}}th harmonic number):

(Euler)

(Negoi)

(Cesàro)

The third formula is also called the Ramanujan expansion.

Exponential

The constant {{math|e{{isup|γ}}}} is important in number theory. Some authors denote this quantity simply as {{math|γ′}}. {{math|e{{isup|γ}}}} equals the following limit, where {{math|pn}} is the {{math|n}}th prime number:

This restates the third of Mertens' theorems.[14] The numerical value of {{math|e{{isup|γ}}}} is:

{{gaps|1.78107|24179|90197|98523|65041|03107|17954|91696|45214|30343|...}} {{OEIS2C|id=A073004}}.

Other infinite products relating to {{math|e{{isup|γ}}}} include:

These products result from the Barnes {{math|G}}-function.

We also have

where the {{math|n}}th factor is the {{math|(n + 1)}}th root of

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.

Continued fraction

The continued fraction expansion of {{mvar|γ}} is of the form {{nowrap|[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...]}} {{OEIS2C|id=A002852}}, which has no apparent pattern. The continued fraction is known to have at least 470,000 terms,[15] and it has infinitely many terms if and only if {{mvar|γ}} is irrational.

Generalizations

Euler's generalized constants are given by

for {{math|0 < α < 1}}, with {{mvar|γ}} as the special case {{math|α {{=}} 1}}.[16] This can be further generalized to

for some arbitrary decreasing function {{math|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:[17][18]

The basic properties are

and if {{math|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 {{mvar>γ}}
Date Decimal digits Author
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 {{val|1050|fmt=gaps}} Helmut Fischer and Karl Zeller
1962 {{val|1271|fmt=gaps}} Donald Knuth
1962 {{val|3566|fmt=gaps}} Dura W. Sweeney
1973 {{val|4879|fmt=gaps}} William A. Beyer and Michael S. Waterman
1977 {{val|20700}} Richard P. Brent
1980 {{val|30100}} Richard P. Brent & Edwin M. McMillan
1993 {{val|172000}} Jonathan Borwein
1999 {{val|108000000}} Patrick Demichel and Xavier Gourdon
2009 {{val|29844489545}} Alexander J. Yee & Raymond Chan[19]
2013 {{val|119,377,958,182}} Alexander J. Yee[19]
2016 {{val|160,000,000,000}} Peter Trueb
2016 {{val|250,000,000,000}} Ron Watkins
2017 {{val|477,511,832,674}} Ron Watkins

Notes

Footnotes
1. ^{{OEIS2C|id=A002852|name=Continued fraction for Euler's constant}}
2. ^{{cite journal| last = Lagarias | first = Jeffrey C. |date=October 2013 | title = Euler's constant: Euler's work and modern developments | journal = Bulletin of the American Mathematical Society | volume = 50 | issue = 4 | page = 556 | url = http://www.ams.org/journals/bull/2013-50-04/S0273-0979-2013-01423-X/S0273-0979-2013-01423-X.pdf | doi=10.1090/s0273-0979-2013-01423-x| arxiv = 1303.1856| bibcode = 1994BAMaS..30..205W }}
3. ^Carl Anton Bretschneider: Theoriae logarithmi integralis lineamenta nova (13 October 1835), Journal für die reine und angewandte Mathematik 17, 1837, pp. 257–285 (in Latin; "{{mvar|γ}} = {{math|c}} = {{nowrap|0,577215 664901 532860 618112 090082 3..}}" on [{{Google |plainurl=yes |id=OAoPAAAAIAAJ |page=260 }} p. 260])
4. ^Augustus De Morgan: The differential and integral calculus, Baldwin and Craddock, London 1836–1842 ("{{mvar|γ}}" on [https://books.google.com/books?id=95x4IrIcHrgC&pg=PA578 p. 578])
5. ^{{cite book|last1=Caves |first1=Carlton M. |author-link1=Carlton M. Caves |last2=Fuchs |first2=Christopher A. |chapter=Quantum information: How much information in a state vector? |title=The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later |year=1996 |isbn=9780750303941 |oclc=36922834 |publisher=Israel Physical Society |arxiv=quant-ph/9601025|bibcode=1996quant.ph..1025C }}
6. ^{{cite journal|last=DeTemple|first=Duane W.|date=May 1993|title=A Quicker Convergence to Euler's Constant|journal=The American Mathematical Monthly|volume=100|issue=5|pages=468–470|doi=10.2307/2324300|issn=0002-9890|jstor=2324300}}
7. ^Havil (2003), pp. 75-78
8. ^Krämer (2005), Blagouchine (2016).
9. ^Vacca (1910), Glaisher (1910), Hardy (1912), Vacca (1925), Kluyver (1927), Krämer (2005), Blagouchine (2016).
10. ^Blagouchine (2014).
11. ^Krämer (2005), Blagouchine (2016), Blagouchine (2018).
12. ^Blagouchine (2016), Alabdulmohsin (2018), pp. 147-148.
13. ^1 Blagouchine (2018).
14. ^http://mathworld.wolfram.com/MertensConstant.html (14)
15. ^1 Havil (2003) p. 97.
16. ^Havil (2003), pp. 117–118
17. ^{{cite journal|first1=M. | last1=Ram Murty | first2=N. | last2=Saradha | title=Euler–Lehmer constants and a conjecture of Erdos| journal = JNT| doi=10.1016/j.jnt.2010.07.004| year=2010|volume=130| issue=12 |pages=2671–2681}}
18. ^{{cite journal| first1=D. H. | last1=Lehmer|year=1975|title=Euler constants for arithmetical progressions | journal=Acta Arith. |volume=27 |number=1| pages=125–142|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf| doi=10.4064/aa-27-1-125-142}}
19. ^1 {{cite web|url=http://www.numberworld.org/nagisa_runs/computations.html|title=Nagisa - Large Computations|author=|date=|website=www.numberworld.org}}

}}
References
  • {{citation |last= Alabdulmohsin |first= Ibrahim M. |title = Summability Calculus. A Comprehensive Theory of Fractional Finite Sums |publisher = Springer-Verlag |isbn = 9783319746487 |year= 2018 }}
  • {{citation |last1= Blagouchine|first1= Iaroslav V.|title= Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results|journal= The Ramanujan Journal|date= 2014|volume= 35|issue= 1|pages= 21–110|ref= harv|doi=10.1007/s11139-013-9528-5}} [https://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten's_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results PDF]
  • {{citation|last1= Blagouchine|first1= Iaroslav V.|title= Expansions of generalized Euler's constants into the series of polynomials in {{math|π−2}} and into the formal enveloping series with rational coefficients only|journal= J. Number Theory |date= 2016|volume= 158|issue= |pages=365–396|arxiv=1501.00740|ref= harv|doi=10.1016/j.jnt.2015.06.012}}
  • {{citation |last1= Blagouchine|first1= Iaroslav V.|title= Three notes on Ser's and Hasse's representations for the zeta-functions |journal= Integers (Electronic Journal of Combinatorial Number Theory) |date= 2018 |volume= 18A|issue= #A3|pages= 1–45|url= http://math.colgate.edu/~integers/vol18a.html|bibcode= 2016arXiv160602044B}} [https://arxiv.org/abs/1606.02044 arXiv]
  • {{cite journal|author1=Borwein, Jonathan M. |author2=David M. Bradley |author3=Richard E. Crandall |title=Computational Strategies for the Riemann Zeta Function

|journal=Journal of Computational and Applied Mathematics
|year=2000
|volume=121
|issue=1–2 |pages=11
|url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf
|doi=10.1016/s0377-0427(00)00336-8|bibcode=2000JCoAM.121..247B}} Derives {{mvar|γ}} as sums over Riemann zeta functions.
  • {{cite journal | last1 = Carl Anton | first1 = Bretschneider | year = 1837 | title = Theoriae logarithmi integralis lineamenta nova | url = | journal = Crelle's Journal | volume = 17 | issue = | pages = 257–285 }} (submitted 1835)
  • {{cite journal|first1=I. |last1=Gerst|title=Some series for Euler's constant

|year=1969|journal=Amer. Math. Monthly|doi=10.2307/2316370|volume=76|issue=3|pages=237–275|jstor=2316370}}
  • {{cite journal|author=James Whitbread Lee Glaisher|year=1872|title=On the history of Euler's constant|journal=Messenger of Mathematics|volume=1|pages=25–30}}, {{JFM|03.0130.01}}
  • {{cite journal|author=James Whitbread Lee Glaisher|year=1910|title=On Dr. Vacca's series for {{mvar|γ}} | journal= Q. J. Pure Appl. Math |volume =41 | pages=365–368}}
  • Gourdon, Xavier, and Sebah, P. (2002) "Collection of formulas for Euler's constant, {{mvar|γ}}."
  • Gourdon, Xavier, and Sebah, P. (2004) "The Euler constant: {{mvar|γ}}."
  • {{cite book |first = Julian |last = Havil |year = 2003 |title = Gamma: Exploring Euler's Constant |publisher = Princeton University Press |isbn = 978-0-691-09983-5 }}
  • {{ cite journal| first1=G. H. |last1= Hardy |title=Note on Dr. Vacca's series for {{mvar|γ}} | journal= Q. J. Pure Appl. Math |volume =43 | pages=215–216 |year=1912}}
  • {{ cite journal| first1=E. A. |last1= Karatsuba |title=Fast evaluation of transcendental functions | journal=Probl. Inf. Transm. |volume =27 | number=44 | pages=339–360 |year=1991}}
  • {{cite journal | last1 = Karatsuba | first1 = E.A. | year = 2000 | title = On the computation of the Euler constant {{mvar|γ}} | url = | journal = Journal of Numerical Algorithms | volume = 24 | issue = 1–2| pages = 83–97 | doi = 10.1023/A:1019137125281 }}
  • {{cite journal | last1 = Kluyver | first1 = J.C. | year = 1927 | title = On certain series of Mr. Hardy | journal = Q. J. Pure Appl. Math. | volume = 50 | pages = 185–192}}
  • Donald Knuth (1997) The Art of Computer Programming, Vol. 1, 3rd ed. Addison-Wesley. {{ISBN|0-201-89683-4}}
  • Krämer, Stefan (2005) Die Eulersche Konstante {{mvar|γ}} und verwandte Zahlen. Ph.D. Thesis, University of Göttingen, Germany.
  • {{cite journal| last = Lagarias | first = Jeffrey C. |date=October 2013 | title = Euler's constant: Euler's work and modern developments | journal = Bulletin of the American Mathematical Society | volume = 50 | issue = 4 | page = 556| doi=10.1090/s0273-0979-2013-01423-x| arxiv = 1303.1856| bibcode = 1994BAMaS..30..205W }}
  • {{cite journal | last1 = Lerch | first1 = M. | year = 1897 | title = Expressions nouvelles de la constante d'Euler | url = | journal = Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften | volume = 42 | issue = | page = 5 }}
  • Lorenzo Mascheroni (1790). "Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur". Galeati, Ticini.
  • {{cite news|first1=Jonathan|last1=Sondow|year=1998 |url=http://home.earthlink.net/~jsondow/id8.html |title= An antisymmetric formula for Euler's constant |journal=Mathematics Magazine |volume=71|pages=219–220}}
  • {{cite journal | last1 = Sondow | first1 = Jonathan | year = 2002 | title = A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant| arxiv = math.NT/0211075| journal = Mathematica Slovaca | volume = 59 | issue = | pages = 307–314 | bibcode = 2002math.....11075S }} with an Appendix by Sergey Zlobin
  • {{cite arXiv | first1=Jonathan | last1= Sondow | year=2003 | eprint=math.CA/0306008 | title= An infinite product for {{math|e{{sup|γ}}}} via hypergeometric formulas for Euler's constant, {{mvar|γ}}}}
  • {{cite news|first=Jonathan|last1= Sondow|year=2003a | arxiv=math.NT/0209070 |title= Criteria for irrationality of Euler's constant| journal=Proceedings of the American Mathematical Society|volume= 131 |pages=3335–3344}}
  • {{cite journal | last1 = Sondow | first1 = Jonathan | year = 2005 | title = Double integrals for Euler's constant and {{math|ln {{sfrac|4|π}}}} and an analog of Hadjicostas's formula | arxiv = math.CA/0211148 | journal = American Mathematical Monthly | volume = 112 | issue = 1| pages = 61–65 | doi=10.2307/30037385| jstor = 30037385 }}
  • {{cite arXiv|first1=Jonathan |last1= Sondow|year=2005| eprint=math.NT/0508042 |title=New Vacca-type rational series for Euler's constant and its 'alternating' analog {{math|ln {{sfrac|4|π}}}}}}
  • {{cite journal| first1=Jonathan |last1=Sondow | first2=Wadim | last2=Zudilin | year=2006 |arxiv=math.NT/0304021 |title= Euler's constant, {{math|q}}-logarithms, and formulas of Ramanujan and Gosper | doi=10.1007/s11139-006-0075-1 | volume=12 |issue=2 | journal=The Ramanujan Journal | pages=225–244}} Ramanujan Journal 12: 225-244.
  • {{cite journal | last1 = Vacca | first1 = G. | year = 1926 | title = Nuova serie per la costante di Eulero, {{mvar|C}} = 0,577...". Rendiconti, Accademia Nazionale dei Lincei, Roma, Classe di Scienze Fisiche | url = | journal = Matematiche e Naturali | volume = 6 | issue = 3| pages = 19–20 }}

External links

  • {{mathworld|urlname=Euler-MascheroniConstant|title=Euler–Mascheroni constant}}
  • Jonathan Sondow.
  • Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
  • Further formulae which make use of the constant: Gourdon and Sebah (2004).
{{DEFAULTSORT:Euler-Mascheroni Constant}}

3 : Mathematical constants|Unsolved problems in mathematics|Leonhard Euler
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