Euler-Mascheroni constant

Euler's constant is a forwarding to this article. For other numbers named after Euler see Euler's numbers (disambiguation) .

${\displaystyle \gamma }$

The Euler-Mascheroni constant (after the mathematicians Leonhard Euler and Lorenzo Mascheroni ), also Euler's constant , is an important mathematical constant that occurs particularly in the areas of number theory and analysis . It is denoted by the Greek letter ( gamma ). ${\displaystyle \gamma }$

The blue area represents Euler's constant.

Its definition is:

${\displaystyle \gamma =\lim _{n\to \infty }\left(H_{n}-\ln n\right)=\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln n\right)=\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,\mathrm {d} x,}$

wherein the n-th harmonic number , the natural logarithm and the rounding function referred to. ${\displaystyle H_{n}}$ ${\displaystyle \ln }$ ${\displaystyle \lfloor x\rfloor }$

Its numerical value is accurate to 100 decimal places (sequence A001620 in OEIS ):

γ = 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 35988 05767 23488 48677 26777 66467 09369 47063 29174 67495 ...

As of May 2020, calculation completed on May 26, 2020, 600,000,000,100 decimal places are known.

General

Despite great efforts, it is still unknown whether this number is rational or irrational , whether it is algebraic or transcendent . It is strongly suspected, however, that it is at least an irrational number. The first concrete attempt to prove this was undertaken in 1926 by Paul Émile Appell with the help of the development by Joseph Ser . By calculating the continued fraction expansion of (sequence A002852 in OEIS ) ${\displaystyle \gamma }$

${\displaystyle \gamma =\left[0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,3,7,1,7,1,1,5,1,49,4,1,65,1,4,7,11,1,399,2,1,3,2,1,2,1,5,3,2,1,\dotsc \right]}$

one obtains lower bounds for positive integers and with (for example 475,006 denominators result in the estimate ). ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle \gamma ={\tfrac {p}{q}}}$ ${\displaystyle q>10^{244.663}}$

In contrast to square roots of rational numbers in the Pythagorean theorem and to the number of circles for the circumference and area of ​​a circle with a rational radius, Euler's constant does not occur in finite elementary geometric problems. However, there are many technical problems that lead to the summation of the finite harmonic row , such as the problem of the center of gravity of the cantilevered arm or the problem of the optimal row elevation in theaters and cinemas. Euler's constant occurs with many problems in analysis , number theory and function theory and especially with special functions . ${\displaystyle \pi }$ ${\displaystyle H_{n}}$

convergence

The existence of Euler's constant results from the telescope sum

${\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n+1)\right)=\sum _{k=1}^{\infty }\left({\frac {1}{k}}-\ln {\frac {k+1}{k}}\right),}$

since is a null sequence , can be used in the defining limit instead of . It applies ${\displaystyle \ln(n+1)-\ln(n)}$ ${\displaystyle \ln(n+1)}$ ${\displaystyle \ln(n)}$

${\displaystyle {\frac {1}{k}}-\ln {\frac {k+1}{k}}={\frac {1}{k}}-\int _{k}^{k+1}{\frac {\mathrm {d} x}{x}}=\int _{k}^{k+1}{\frac {x-k}{xk}}\,\mathrm {d} x={\frac {1}{k}}\int _{0}^{1}{\frac {x}{x+k}}\,\mathrm {d} x.}$

Because of

${\displaystyle {\frac {1}{2(k+1)}}=\int _{0}^{1}{\frac {x}{1+k}}\,\mathrm {d} x\leq \int _{0}^{1}{\frac {x}{x+k}}\,\mathrm {d} x\leq \int _{0}^{1}{\frac {x}{k}}\,\mathrm {d} x={\frac {1}{2k}}}$

so it applies

${\displaystyle {\frac {1}{2k\cdot (k+1)}}\leq {\frac {1}{k}}-\ln {\frac {k+1}{k}}\leq {\frac {1}{2k^{2}}}}$

and thus the sum converges according to the majorant criterion .

In particular it follows from this elementary argument and

${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k\cdot (k+1)}}=\sum _{k=1}^{\infty }\left({\frac {1}{k}}-{\frac {1}{k+1}}\right)=1}$

as well as the Basel problem that

${\displaystyle {\frac {1}{2}}\leq \gamma \leq {\frac {\pi ^{2}}{12}}}$

applies.

The Euler-Mascheroni constant in mathematical problems

Euler's constant occurs frequently in mathematics and sometimes quite unexpectedly in different sub-areas. It mainly occurs in limit value processes of sequences of numbers and functions as well as in limit values ​​in differential and integral calculus . The occurrence can (as with other mathematical constants ) be subdivided depending on the type of limit value as follows:

1. As a function value or limit value of special functions .

The value is the negative of the derivative of the gamma function at position 1, i.e. ${\displaystyle \gamma }$

${\displaystyle \Gamma ^{\prime }(1)=\psi (1)=-\gamma }$.

This results in the following limit value representations, whereby the Riemann zeta function and the digamma function denote: ${\displaystyle \zeta (s)}$ ${\displaystyle \psi (z)}$

${\displaystyle \lim _{s\to 1}\left(\zeta (s)-{\frac {1}{s-1}}\right)=\gamma }$

${\displaystyle \lim _{z\to 0}\left\{\Gamma (z)-{\frac {1}{z}}\right\}=\lim _{z\to 0}\left\{\psi (z)+{\frac {1}{z}}\right\}=-\gamma }$

${\displaystyle \lim _{z\to 0}{\frac {1}{z}}\left\{{\frac {1}{\Gamma (1+z)}}-{\frac {1}{\Gamma (1-z)}}\right\}=2\gamma }$

${\displaystyle \lim _{z\to 0}{\frac {1}{z}}\left\{{\frac {1}{\psi (1-z)}}-{\frac {1}{\psi (1+z)}}\right\}={\frac {\pi ^{2}}{3\gamma ^{2}}}}$

2. In developments of special functions, e.g. B. in the series expansion of the integral logarithm by Leopold Schendel , the Bessel functions or the Weierstrass ' representation of the gamma function.

3. When evaluating certain integrals.

There is an abundance here, for example:

${\displaystyle {\begin{aligned}\gamma &=-\int _{0}^{1}\ln(-\ln x)\,\mathrm {d} x\\\gamma &=-\int _{0}^{\infty }e^{-x}\ln x\,\mathrm {d} x\\\gamma &=\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{xe^{x}}}\right)\,\mathrm {d} x\\\gamma &=\int _{0}^{1}\left({\frac {1}{\ln x}}+{\frac {1}{1-x}}\right)\,\mathrm {d} x\\\gamma &={\frac {1}{2}}+2\int _{0}^{\infty }{\frac {\sin(\arctan x)}{(e^{2\pi x}-1){\sqrt {1+x^{2}}}}}\,\mathrm {d} x\end{aligned}}}$

or

${\displaystyle {\begin{aligned}\int _{0}^{\infty }e^{-x}\ln ^{2}x\,\mathrm {d} x&={\frac {\pi ^{2}}{6}}+\gamma ^{2}\\\int _{0}^{\infty }e^{-x^{2}}\ln x\,\mathrm {d} x&=-{\frac {\sqrt {\pi }}{4}}(\gamma +2\ln 2)\end{aligned}}}$

There are also many invariant parameter integrals, e.g. B .:

${\displaystyle {\begin{aligned}\gamma &=\int _{0}^{\infty }\left({\frac {1}{x^{k}+1}}-e^{-x}\right){\frac {\mathrm {d} x}{x}},\quad k>0\\\gamma &=\int _{0}^{\infty }\left({\frac {1}{kx+1}}-e^{-kx}\right){\frac {\mathrm {d} x}{x}},\quad k>0\end{aligned}}}$

One can also express it as a double integral (J. Sondow 2003, 2005) with the equivalent series: ${\displaystyle \gamma }$

${\displaystyle \gamma =\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-xy)\ln(xy)}}\,\mathrm {d} x\,\mathrm {d} y=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right)}$.

There is an interesting comparison (J. Sondow 2005) of the double integral and the alternating series:

${\displaystyle \ln \left({\frac {4}{\pi }}\right)=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1+xy)\ln(xy)}}\,\mathrm {d} x\,\mathrm {d} y=\sum _{n=1}^{\infty }(-1)^{n-1}\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right)}$.

In this sense one can say that the "alternating Euler's constant" is (sequence A094640 in OEIS ). ${\displaystyle \ln {\big (}{\tfrac {4}{\pi }}{\big )}}$

Also, these two are constants with the pair

${\displaystyle \sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}=\gamma ,}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}}=\ln \left({\frac {4}{\pi }}\right)}$

linked by series, where and are the number of ones or zeros in the binary expansion of (Sondow 2010). ${\displaystyle N_{1}(n)}$ ${\displaystyle N_{0}(n)}$ ${\displaystyle n}$

There is also an equally rich abundance of infinite sums and products, for example

${\displaystyle {\begin{aligned}e^{\gamma }&=\lim _{n\to \infty }{\frac {1}{\ln n}}\prod _{p\leq n,p{\text{ prim}}}\left(1-{\frac {1}{p}}\right)^{-1}\\{\frac {6}{\pi ^{2}}}e^{\gamma }&=\lim _{n\to \infty }{\frac {1}{\ln n}}\prod _{p\leq n,p{\text{ prim}}}\left(1+{\frac {1}{p}}\right)\\\gamma &=\lim _{x\to 1^{+}}\sum _{n=1}^{\infty }\left({\frac {1}{n^{x}}}-{\frac {1}{x^{n}}}\right).\end{aligned}}}$

The last formula was found in 1998 by Sondow.

4. As a limit of series . The simplest example results from the limit value definition:

${\displaystyle \gamma =\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln {\frac {n+1}{n}}\right)}$.

Series with rational terms are from Euler, Fontana and Mascheroni, Giovanni Enrico Eugenio Vacca , S. Ramanujan and Joseph Ser . There are innumerable variations of series with irrational terms, the terms of which consist of rationally weighted values ​​of the Riemannian zeta function at the odd argument positions ζ (3), ζ (5),…. An example of a particularly rapidly converging series is:

${\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n+1)-1}{(2n+1)2^{2n}}}=1+\ln 2-\ln 3-\gamma =}$ 0.0173192269903 ...

Another series results from the Kummer series of the gamma function:

${\displaystyle \gamma =\ln \pi -4\ln \Gamma {\big (}{\tfrac {3}{4}}{\big )}+{\frac {4}{\pi }}\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {\ln(2k+1)}{2k+1}}}$

Designations

One can say that Euler's constant is the constant with the most names. Euler himself called them C and occasionally O or n . However, it is doubtful whether he wanted to introduce an independent symbol for his constant. Mascheroni did not designate the constant with γ - as often claimed - but with A. The γ misunderstanding stems from the often unreviewed article by JWL Glaisher (where Glaisher expressly states that he has not seen Mascheroni's book):

“Euler's constant (which throughout this note will be called γ after Mascheroni, De Morgan, & c.) […]
It is clearly convenient that the constant should generally be denoted by the same letter. Euler used C and O for it; Legendre, Lindman, & c., C ; De Haan A ; and Mascheroni, De Morgan, Boole, & c., have written it γ , which is clearly the most suitable, if it is to have a distinctive letter assigned to it. It has sometimes (as in Crelle, t. 57, p. 128) been quoted as Mascheroni's constant, but it is evident that Euler's labors have abundantly justified his claim to its being named after him. "

- JWL Glaisher : On the history of Euler's constant, 1872, pp. 25 and 30

Other mathematicians use the terms C , c , ℭ, H , γ , E , K , M , l . The origin of today's common name γ is not certain. Carl Anton Bretschneider used the term γ next to c in an article written in 1835 and published in 1837, Augustus De Morgan introduced the term γ in a textbook published in parts from 1836 to 1842 as part of the treatment of the gamma function.

Generalizations

Euler's constant knows several generalizations. The most important and best known is that of the Stieltjes constants :

${\displaystyle \gamma _{n}:=\lim _{N\to \infty }\left(\sum _{k=1}^{N}{\frac {\log ^{n}k}{k}}-{\frac {\log ^{n+1}N}{n+1}}\right),\quad n=0,1,2,\dotsc }$

Number of calculated decimal places

In 1734 Leonhard Euler calculated six decimal places (five valid), later 16 places (15 valid). In 1790 Lorenzo Mascheroni calculated 32 decimal places (30 valid), of which the three places 20 to 22 are wrong - apparently due to a typographical error, but they are given several times in the book. The error prompted several recalculations.

Number of published valid decimal places of γ

date	Put	author
1734	5	Leonhard Euler
1735	15th	Leonhard Euler
1790	19th	Lorenzo Mascheroni
1809	22nd	Johann Georg Soldner
1811	22nd	Carl Friedrich Gauss
1812	40	Friedrich Bernhard Gottfried Nicolai
1826	19th	Adrien-Marie Legendre
1857	34	Christian Fredrik Lindman
1861	41	Ludwig Oettinger
1867	49	William Shanks
1871	99	JWL Glaisher
1871	101	William Shanks
1877	262	John Couch Adams
1952	328	John William Wrench, Jr.
1961	1,050	Helmut Fischer & Karl Zeller
1962	1,270	Donald E. Knuth
1962	3,566	Dura W. Sweeney
1973	4,879	William A. Beyer & Michael S. Waterman
1976	20,700	Richard P. Brent
1979	30,100	Richard P. Brent & Edwin M. McMillan
1993	172,000	Jonathan Borwein
1997	1,000,000	Thomas Papanikolaou
1998	7,286,255	Xavier Gourdon
1999	108,000,000	Xavier Gourdon & Patrick Demichel
December 8, 2006	116.580.041	Alexander J. Yee & Raymond Chan
January 18, 2009	14,922,244,771	Alexander J. Yee & Raymond Chan
March 13, 2009	29,844,489,545	Alexander J. Yee & Raymond Chan
December 22, 2013	119.377.958.182	Alexander J. Yee
15th March 2016	160,000,000,000	Peter Trueb
May 18, 2016	250,000,000,000	Ron Watkins
23rd August 2017	477,511,832,674	Ron Watkins
May 26, 2020	600,000,000,100	Seungmin Kim & Ian Cutress

See also

• Meissel-Mertens constant - prime analogue of the Euler-Mascheroni constant

literature

• M. Lerch : Expressions nouvelles de la constante d'Euler (July 9, 1897), session reports of the royal. Bohemian Society of Sciences. Mathematical and natural science class, 1897, XLII pp. 1–5 (French; yearbook report )
• Jonathan Sondow: Criteria for irrationality of Euler's constant (June 4, 2002), Proceedings of the AMS 131, November 2003, pp. 3335–3344 (English; Zentralblatt report )
• Steven R. Finch: Euler-Mascheroni constant, γ, Chapter 1.5 in Mathematical constants, Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , pp. 28-40 (English; Zentralblatt review ; Finch's website for the book with errata and addenda: Mathematical Constants )
• Julian Havil : Gamma: Euler's constant, prime number beaches and the Riemann hypothesis . Springer, Berlin 2007, ISBN 978-3-540-48495-0 ( Zentralblatt review )
• Thomas P. Dence, Joseph B. Dence: A survey of Euler's constant ( PDF file, 432 kB), Mathematics Magazine 82, October 2009, pp. 255–265 (English)
• Jeffrey C. Lagarias : Euler's constant: Euler's work and modern developments, Bulletin AMS, Volume 50, 2013, pp. 527–628, arxiv : 1303.1856 [math.NT], 2013 (English)

Web links

• Eric W. Weisstein : Euler-Mascheroni Constant . In: MathWorld (English).
• Episode A053977 in OEIS (Engel development of γ)
• Episode A006284 in OEIS (Pierce expansion of γ)
• Xavier Gourdon, Pascal Sebah: The Euler constant: γ auf Numbers, constants and computation, April 14, 2004 (English)

Individual evidence

1. ↑ ^{a b c d e f g h} Records set by y-cruncher by Alexander Yee, accessed on May 28, 2020 (English)
2. ^ Thomas Jagau: Is the Euler-Mascheroni constant an irrational number? , Question of the Week, May 11, 2011 in JunQ: Journal of Unresolved Questions
3. ↑ Bruno Haible, Thomas Papanikolaou: Fast multiprecision evaluation of series of rational numbers ( PDF file, 197 kB), 1997 (English)
4. ↑ Jonathan Sondow: Criteria for irrationality of Euler's constant , Proceedings of the American Mathematical Society 131, 2003, pp. 3335-3344 (English)
5. ↑ Jonathan Sondow: Double integrals for Euler's constant and ln 4 / π and an analog of Hadjicostas's formula , American Mathematical Monthly 112, 2005, pp. 61–65 (English)
6. ↑ Jonathan Sondow:  New Vacca-type rational series for Euler's constant and its “alternating” analog ln 4 / π ,  Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson, Springer, New York, 2010, p. 331–340 (English)
7. ↑ Jonathan Sondow: An antisymmetric formula for Euler's constant  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.   ( PDF file, 227 kB), Mathematics Magazine 71, June 1998, pp. 219–220 (English;)
8. ↑ JWL Glaisher : On the history of Euler's constant . The Messenger of Mathematics 1, 1872, pp. 25–30 (English)
9. ^ Car. Ant Bretschneider. : Theoriae logarithmic integralis lineamenta nova (13 October 1835). Journal for pure and applied mathematics 17, 1837, pp. 257–285 (Latin; " γ = c = 0.577215 664901 532860 618112 090082 3 ..." on p. 260 )
10. ^ Augustus De Morgan : The differential and integral calculus , Baldwin and Craddock, London 1836–1842 (English; "γ" on p. 578 )
11. ↑ Leonh. Eulero : De progressionibus harmonicis observationes (March 11, 1734), Commentarii academiae scientiarum imperialis Petropolitanae 7, 1740, pp. 150–161 (Latin; "C = 0.577218" on p. 157 )
12. ↑ Leonh. Eulero : Inventio summae cuiusque seriei ex dato termino generali (October 13, 1735), Commentarii academiae scientiarum imperialis Petropolitanae 8, 1741, pp. 9–22 (Latin; "constans illa addita = 0.5772156649015329" on p. 19 )
13. ↑ Laurentio Mascheronio : Adnotationes ad calculum integralem Euleri / In quibus nonnulla Problemata ab Eulero proposita resolvuntur . Petrus Galeatius, Ticini 1790–1792 (Latin; "A = 0.577215 664901 532860 618112 090082 39" on p. 23 and p. 45 )
14. ↑ J. Soldner : Théorie et tables d'une nouvelle fonction transcendante , Lindauer, Munich 1809 (French; " H = 0.5772156649015328606065" on p. 13 )
15. ↑ ^{a b} Carolus Fridericus Gauss : Disquisitiones generales circa seriem infinitam Pars I (January 30, 1812), Commentationes Societatis Regiae Scientiarum Gottingensis recentiores 2 (classis mathematicae), 1813, pp. 3–46 (Latin; “ψ ₀ = −0, 5772156649 0153286060 653 "and" ψ ₀ = −0.5772156649 0153286060 6512090082 4024310421 "on p. 36 ; also in Gauß: Werke. Volume 3, p. 154 )
16. ↑ AM Legendre : Traité des fonctions elliptiques et des intégrales Eulériennes (Volume 2), Huzard-Courcier, Paris 1826, p. 434 (French)
17. ↑ Christiano Fr. Lindman : De vero valore constantis, quae in logarithmo integrali occurit, Archive of Mathematics and Physics 29, 1857, pp. 238–240 (Latin)
18. ↑ L. Oettinger : About the correct determination of the value of the constants of the integral logarithm (October 1861), Journal for pure and applied mathematics 60, 1862, pp. 375–376
19. ^ William Shanks : On the calculation of the numerical value of Euler's constant, which Professor Price, of Oxford, calls E (March 28 / April 9 / August 29, 1867 / June 14, 1869), Proceedings of the Royal Society of London 15, 1867, pp. 429-431 431-432 ; 16, 1868, p. 154 ; 18, 1870, p. 49 (English)
20. ↑ JWL Glaisher : On the calculation of Euler's constant (June 6/14, 1871), Proceedings of the Royal Society of London 19, 1871, pp. 514-524 (English)
21. ^ William Shanks : On the numerical value of Euler's constant, and on the summation of the harmonic series employed in obtaining such value (August 30, 1871), Proceedings of the Royal Society of London 20, 1872, pp. 29-34 (English )
22. ↑ JC Adams : Note on the value of Euler's constant; likewise on the values ​​of the Napierian logarithms of 2, 3, 5, 7, and 10, and of the modulus of common logarithms, all carried to 260 places of decimals (December 6, 1877), Proceedings of the Royal Society of London 27 , 1878, pp. 88–94 (English)
23. ↑ JW Wrench, Jr .: Note 141. A new calculation of Euler's constant, Mathematical tables and other aids to computation 6, October 1952, p. 255 (English)
24. ↑ H. Fischer, K. Zeller : Bernoullische Numbers and Euler's Constant, Journal for Applied Mathematics and Mechanics 41 (special issue), 1961, pp. T71 – T72 ( Zentralblatt summary )
25. ↑ Donald E. Knuth : Euler's constant to 1271 places (January 12, 1962), Mathematics of Computation 16, 1962, pp. 275–281 (English)
26. ↑ Dura W. Sweeney: On the computation of Euler's constant (June 29, 1962), Mathematics of Computation 17, 1963, pp. 170-178 (English)
27. ↑ WA Beyer, MS Waterman : Error analysis of a computation of Euler's constant (June 26, 1973), Mathematics of Computation 28, 1974, pp. 599-604 (English)
28. ↑ Richard P. Brent : Computation of the regular continued fraction for Euler's constant (September 27, 1976), Mathematics of Computation 31, 1977, pp. 771-777 (English)
29. ↑ Richard P. Brent, Edwin M. McMillan : Some new algorithms for high-precision computation of Euler's constant (January 22 / May 15, 1979), Mathematics of Computation 34, 1980, pp. 305-312 (English)
30. ↑ Alexander Jih-Hing Yee: Euler-Mascheroni Constant - 116 million digits on a laptop. Retrieved March 20, 2020 .