Euler's constant
Euler's constant | |
---|---|
γ 0.57721...[1] | |
General information | |
Type | Unknown |
Fields | |
History | |
Discovered | 1734 |
By | Leonhard Euler |
First mention | De Progressionibus harmonicis observationes |
Named after |
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]
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]- The Weierstrass product formula for the gamma function and the Barnes G-function.[8][9]
- The asymptotic expansion of the gamma function, .
- Evaluations of the digamma function at rational values.[10]
- The Laurent series expansion for the Riemann zeta function*, where it is the first of the Stieltjes constants.[11]
- Values of the derivative of the Riemann zeta function and Dirichlet beta function.[12]: 137 [13]
- In connection to the Laplace and Mellin transform.[14][15]
- In the regularization/renormalization of the harmonic series as a finite value.
- Expressions involving the exponential and logarithmic integral.*[16][17]
- A definition of the cosine integral.*[16]
- In relation to Bessel functions.[18][19][20][21]
- Asymptotic expansions of modified Struve functions.[22]
- In relation other special functions.[23][24]
Number theory
[edit]- An inequality for Euler's totient function.[25]
- The growth rate of the divisor function.[26]
- A formulation of the Riemann hypothesis.[27][28]
- The third of Mertens' theorems.*[29]
- The calculation of the Meissel–Mertens constant.[30]
- Lower bounds to specific prime gaps.[31]
- An approximation of the average number of divisors of all numbers from 1 to a given n.[32]
- The Lenstra–Pomerance–Wagstaff conjecture on the frequency of Mersenne primes.[33]
- An estimation of the efficiency of the euclidean algorithm.[34]
- Sums involving the Möbius and von Mangolt function.
In other fields
[edit]- In some formulations of Zipf's law.
- The answer to the coupon collector's problem.*
- The mean of the Gumbel distribution.
- An approximation of the Landau distribution.
- The information entropy of the Weibull and Lévy distributions, and, implicitly, of the chi-squared distribution for one or two degrees of freedom.
- An upper bound on Shannon entropy in quantum information theory.[35]
- In dimensional regularization of Feynman diagrams in quantum field theory.
- In the BCS equation on the critical temperature in BCS theory of superconductivity.*
- Fisher–Orr model for genetics of adaptation in evolutionary biology.[36]
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]
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 is transcendental, where and are the usual 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, if 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 showed that the generalized Euler constants have the same property, [3][44] [45] where the generalized Euler constant are defined as where is a fixed list of prime numbers, if at least one of the primes in is a prime factor of , and otherwise. In particular, .
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.
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 Malmsten–Kummer 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]
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]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 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]- ^ Jump up to: 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.
- ^ Jump up to: a b c d Weisstein, Eric W. "Euler-Mascheroni Constant". mathworld.wolfram.com. Retrieved 2024-10-19.
- ^ Jump up to: a b c d e f g h i j k l m n o p q r Lagarias 2013.
- ^ Bretschneider 1837, "γ = c = 0,5772156649015328606181120900823..." on p. 260.
- ^ De Morgan, Augustus (1836–1842). The differential and integral calculus. London: Baldwin and Craddoc. "γ" on p. 578.
- ^ Brent, Richard P. (1994). "Ramanujan and Euler's Constant" (PDF). Proc. Symp. Applied Math. Proceedings of Symposia in Applied Mathematics. 48: 541–545. doi:10.1090/psapm/048/1314887. ISBN 978-0-8218-0291-5.
- ^ Sondow, Jonathan (2004). "The Euler constant: γ". Retrieved 2024-11-01.
- ^ 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.
- ^ "DLMF: §5.17 Barnes' 𝐺-Function (Double Gamma Function) ‣ Properties ‣ Chapter 5 Gamma Function". dlmf.nist.gov. Retrieved 2024-11-01.
- ^ Weisstein, Eric W. "Digamma Function". mathworld.wolfram.com. Retrieved 2024-10-30.
- ^ Weisstein, Eric W. "Stieltjes Constants". mathworld.wolfram.com. Retrieved 2024-11-01.
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- ^ Jump up to: 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.
- ^ Williams, John (1973). Laplace transforms. Problem solvers. London: Allen & Unwin. ISBN 978-0-04-512021-5.
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- ^ Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers". Illinois Journal of Mathematics. 6 (1): 64–94. doi:10.1215/ijm/1255631807. ISSN 0019-2082.
- ^ Hardy, Godfrey H.; Wright, Edward M.; Silverman, Joseph H. (2008). Heath-Brown, D. R. (ed.). An introduction to the theory of numbers. Oxford mathematics (6th ed.). Oxford New York Auckland: Oxford University Press. p. 469-471. ISBN 978-0-19-921986-5.
- ^ Jump up to: a b Waldschmidt, Michel (2023). "On Euler's Constant" (PDF). Sorbonne Université, Institut de Mathématiques de Jussieu, Paris.
- ^ Robin, Guy (1984). "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann" (PDF). Journal de mathématiques pures et appliquées. 63: 187–213.
- ^ Jump up to: a b Weisstein, Eric W. "Mertens Theorem". mathworld.wolfram.com. Retrieved 2024-10-08.
- ^ Weisstein, Eric W. "Mertens Constant". mathworld.wolfram.com. Retrieved 2024-11-01.
- ^ Pintz, János (1997-04-01). "Very Large Gaps between Consecutive Primes". Journal of Number Theory. 63 (2): 286–301. doi:10.1006/jnth.1997.2081. ISSN 0022-314X.
- ^ Jump up to: a b Conway, John H.; Guy, Richard (1998-03-16). The Book of Numbers. Springer Science & Business Media. ISBN 978-0-387-97993-9.
- ^ "Heuristics: Deriving the Wagstaff Mersenne Conjecture". t5k.org. Retrieved 2024-11-01.
- ^ Weisstein, Eric W. "Porter's Constant". mathworld.wolfram.com. Retrieved 2024-11-01.
- ^ 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.
- ^ 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.
- ^ "Eulers Constant". num.math.uni-goettingen.de. Retrieved 2024-10-19.
- ^ Waldschmidt, Michel (2023). "Some of the most famous open problems in number theory" (PDF).
- ^ Jump up to: 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.
- ^ Aptekarev, A. I. (28 February 2009). "On linear forms containing the Euler constant". arXiv:0902.1768 [math.NT].
- ^ 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.
- ^ Mahler, Kurt (4 June 1968). "Applications of a theorem by A. B. Shidlovski" (PDF). 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.
- ^ Jump up to: 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.
- ^ 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.
- ^ Diamond, H. G.; Ford, K. (2008). "Generalized Euler constants". Mathematical Proceedings of the Cambridge Philosophical Society. 145 (1). Cambridge University Press: 27–41. arXiv:math/0703508. Bibcode:2008MPCPS.145...27D. doi:10.1017/S0305004108001187.
- ^ 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.
- ^ Papanikolaou, T. (1997). Entwurf und Entwicklung einer objektorientierten Bibliothek für algorithmische Zahlentheorie (Thesis) (in German). Universität des Saarlandes.
- ^ Weisstein, Eric W. "Euler-Mascheroni Constant Digits". mathworld.wolfram.com. Retrieved 2024-10-19.
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- ^ Pilehrood, Khodabakhsh Hessami; Pilehrood, Tatiana Hessami (2013-12-29), On a continued fraction expansion for Euler's constant, arXiv:1010.1420
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- ^ Jump up to: a b c d Krämer, Stefan (2005). Die Eulersche Konstante γ und verwandte Zahlen (in German). University of Göttingen.
- ^ 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. - ^ 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.
- ^ Jump up to: 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. Bibcode:2008RvMad.102..199B. doi:10.1007/BF03191819.
γ/2 in (10) reflects the residual (finite part) of ζ(1)/2, of course.
See formulas 1 and 10. - ^ 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.
- ^ 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.
- ^ Lodge, A. (1904). "An approximate expression for the value of 1 + 1/2 + 1/3 + ... + 1/r". Messenger of Mathematics. 30: 103–107.
- ^ 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.
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- ^ Lagarias 2013, p. 13.
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- ^ Sondow, Jonathan (1 August 2005a). New Vacca-type rational series for Euler's constant and its 'alternating' analog . arXiv:math.NT/0508042.
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- ^ Pilehrood, Khodabakhsh Hessami; Pilehrood, Tatiana Hessami (2008-08-04), Vacca-type series for values of the generalized-Euler-constant function and its derivative, arXiv:0808.0410
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Further reading
[edit]- Borwein, Jonathan M.; David M. Bradley; Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function". Journal of Computational and Applied Mathematics. 121 (1–2): 11. Bibcode:2000JCoAM.121..247B. doi:10.1016/s0377-0427(00)00336-8. Derives γ as sums over Riemann zeta functions.
- Finch, Steven R. (2003). Mathematical Constants. Encyclopedia of Mathematics and its Applications. Vol. 94. Cambridge: Cambridge University Press. ISBN 0-521-81805-2.
- Gerst, I. (1969). "Some series for Euler's constant". Amer. Math. Monthly. 76 (3): 237–275. doi:10.2307/2316370. JSTOR 2316370.
- Glaisher, James Whitbread Lee (1872). "On the history of Euler's constant". Messenger of Mathematics. 1: 25–30. JFM 03.0130.01.
- Gourdon, Xavier; Sebah, P. (2002). "Collection of formulae for Euler's constant, γ".
- Gourdon, Xavier; Sebah, P. (2004). "The Euler constant: γ".
- Julian Havil (2003): GAMMA: Exploring Euler's Constant, Princeton University Press, ISBN 978-0-69114133-6.
- Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (44): 339–360.
- Karatsuba, E.A. (2000). "On the computation of the Euler constant γ". Journal of Numerical Algorithms. 24 (1–2): 83–97. doi:10.1023/A:1019137125281. S2CID 21545868.
- Knuth, Donald (1997). The Art of Computer Programming, Vol. 1 (3rd ed.). Addison-Wesley. pp. 75, 107, 114, 619–620. ISBN 0-201-89683-4.
- Lehmer, D. H. (1975). "Euler constants for arithmetical progressions" (PDF). Acta Arith. 27 (1): 125–142. doi:10.4064/aa-27-1-125-142.
- Lerch, M. (1897). "Expressions nouvelles de la constante d'Euler". Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften. 42: 5.
- Mascheroni, Lorenzo (1790). Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur. Galeati, Ticini.
- Sondow, Jonathan (2002). "A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant". Mathematica Slovaca. 59: 307–314. arXiv:math.NT/0211075. Bibcode:2002math.....11075S. doi:10.2478/s12175-009-0127-2. S2CID 16340929. with an Appendix by Sergey Zlobin
External links
[edit]- "Euler constant". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
- Weisstein, Eric W. "Euler–Mascheroni constant". MathWorld.
- Jonathan Sondow.
- Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
- Further formulae which make use of the constant: Gourdon and Sebah (2004).