Euler's constant explained

Euler's constant
Type:	Unknown
Approximation:	0.57721...
Discovery Date:	1734
Discovery Person:	Leonhard Euler
Discovery Work:	De Progressionibus harmonicis observationes

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

Here, represents the floor function.

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

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 and for the constant. In 1790, the Italian mathematician Lorenzo Mascheroni used the notations and for the constant. The notation 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. For example, the German mathematician Carl Anton Bretschneider used the notation in 1835, and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.

Appearances

Euler's constant appears, among other places, in the following (where '*' 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
• The asymptotic expansion of the gamma function for small arguments.
• 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
• Fisher–Orr model for genetics of adaptation in evolutionary biology
• Bardeen–Cooper–Schrieffer theory of superconductivity (BCS theory), where it appears as prefactor

2 e ^{\gamma/\pi=1.134} in the BCS equation on the critical temperature.

Properties

The number has not been proved algebraic or transcendental. In fact, it is not even known whether is irrational. Using a continued fraction analysis, Papanikolaou showed in 1997 that if is rational, its denominator must be greater than 10²⁴⁴⁶⁶³. The ubiquity of revealed by the large number of equations below makes the irrationality of a major open question in mathematics.

However, some progress has been made. Kurt Mahler showed in 1968 that the number

(\pi)/(2) (Y _{0(2)})/(J _{0(2)}) - \gamma is transcendental (here, J _{\alpha(x)} and Y _{\alpha(x)} are Bessel functions). In 2009 Alexander Aptekarev proved that at least one of Euler's constant and the Euler–Gompertz constant is irrational; Tanguy Rivoal proved in 2012 that at least one of them is transcendental. In 2010 M. Ram Murty and N. Saradha showed that at most one of the numbers of the form

$$\backslash gamma(a,q)\; =\; \backslash lim\_\backslash left(\backslash left(\backslash sum\_^n\backslash right)\; -\; \backslash frac\backslash right)$$

with and is algebraic; this family includes the special case . In 2013 M. Ram Murty and A. Zaytseva found a different family containing, which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.

Relation to gamma function

is related to the digamma function, and hence the derivative of the gamma function, when both functions are evaluated at 1. Thus:

$$-\backslash gamma\; =\; \backslash Gamma\text{'}(1)\; =\; \backslash Psi(1).$$

This is equal to the limits:

Further limit results are:

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

Relation to the zeta function

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

The constant

\gamma can also be expressed in terms of the sum of the reciprocals of non-trivial zeros \rho of the zeta function:^[1] \gamma = log 4 \pi + \sum _{\rho} (2)/(\rho) - 2 Other series related to the zeta function include:

The error term in the last equation is a rapidly decreasing function of . 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:

and the following formula, established in 1898 by de la Vallée-Poussin:

$$\backslash gamma\; =\; \backslash lim\_\backslash frac1\backslash ,\; \backslash sum\_^n\; \backslash left(\backslash left\backslash lceil\; \backslash frac\; \backslash right\backslash rceil\; -\; \backslash frac\backslash right)$$

where are ceiling brackets.This formula indicates that when taking any positive integer and dividing it by each positive integer less than, the average fraction by which the quotient falls short of the next integer tends to (rather than 0.5) as 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:

$$\backslash gamma\; =\backslash lim\_\backslash left(\backslash sum\_^n\; \backslash frac1\; -\; \backslash log\; n\; -\backslash sum\_^\backslash infty\; \backslash frac\backslash right),$$

where is the Hurwitz zeta function. The sum in this equation involves the harmonic numbers, . Expanding some of the terms in the Hurwitz zeta function gives:

$$H\_n\; =\; \backslash log(n)\; +\; \backslash gamma\; +\; \backslash frac1\; -\; \backslash frac1\; +\; \backslash frac1\; -\; \backslash varepsilon,$$where

can also be expressed as follows where is the Glaisher–Kinkelin constant:

$$\backslash gamma\; =12\backslash ,\backslash log(A)-\backslash log(2\backslash pi)+\backslash frac\backslash ,\backslash zeta\text{'}(2)$$

can also be expressed as follows, which can be proven by expressing the zeta function as a Laurent series:

$$\backslash gamma=\backslash lim\_\backslash left(-n+\backslash zeta\backslash left(\backslash frac\backslash right)\backslash right)$$

Relation to triangular numbers

Numerous formulations have been derived that express

\gamma in terms of sums and logarithms of triangular numbers.^{[2] [3] [4] [5]} One of the earliest of these is a formula^{[6] [7]} for the harmonic number attributed to Srinivasa Ramanujan where \gamma is related to style ln 2 T _{k} in a series that considers the powers of style (1)/(T _{k}) (an earlier, less-generalizable proof^{[8] [9]} by Ernesto Cesàro gives the first two terms of the series, with an error term):\begin{align} \gamma & = H _{u} - (1)/(2) ln 2 T _{u} - \sum ^{v} _{k =1} (R(k))/(T ^{k} _{u}) - \Theta _{v} (R(v +1 ))/(T ^{v +1} _{u}) \end{align}

From Stirling's approximation^{[2] [10]} follows a similar series:

\gamma = ln 2 \pi - \sum ^{n} _{k =2} (\zeta(k))/(T _{k})

The series of inverse triangular numbers also features in the study of the Basel problem^[11] posed by Pietro Mengoli. Mengoli proved that

style \sum ^{inf ty} _{k =1} (1)/(2 T _{k}) = 1 , a result Jacob Bernoulli later used to estimate the value of \zeta( 2 ) , placing it between 1 and style \sum ^{inf ty} _{k =1} (2)/(2 T _{k}) = \sum ^{inf ty} _{k =1} (1)/(T _{k}) = 2 . This identity appears in a formula used by Bernhard Riemann to compute roots of the zeta function,^[12] where \gamma is expressed in terms of the sum of roots \rho plus the difference between Boya's expansion and the series of exact unit fractions style \sum ^{n} _{k =1} (1)/(T _{k}) :\gamma - ln 2 = ln 2 \pi + \sum _{\rho} (2)/(\rho) - \sum ^{n} _{k =1} (1)/(T _{k})

Integrals

equals the value of a number of definite integrals:

where is the fractional harmonic number, and

\{ 1 /x\} is the fractional part of 1 /x .

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 .

Definite integrals in which appears include:

One can express using a special case of Hadjicostas's formula as a double integral with equivalent series:

An interesting comparison by Sondow is the double integral and alternating series

It shows that may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series

where and are the number of 1s and 0s, respectively, in the base 2 expansion of .

We also have Catalan's 1875 integral

$$\backslash gamma\; =\; \backslash int\_0^1\; \backslash left(\backslash frac1\backslash sum\_^\backslash infty\; x^\backslash right)\backslash ,dx.$$

Series expansions

In general,

$$\backslash gamma\; =\; \backslash lim\_\backslash left(\backslash frac+\backslash frac+\backslash frac\; +\; \backslash ldots\; +\; \backslash frac\; -\; \backslash log(n+\backslash alpha)\; \backslash right)\; \backslash equiv\; \backslash lim\_\; \backslash gamma\_n(\backslash alpha)$$

for any . However, the rate of convergence of this expansion depends significantly on . In particular, exhibits much more rapid convergence than the conventional expansion . 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 :$$\backslash gamma\; =\; \backslash sum\_^\backslash infty\; \backslash left(\backslash frac\; 1\; k\; -\; \backslash log\backslash left(1+\backslash frac\; 1\; k\; \backslash right)\backslash right).$$

The series for is equivalent to a series Nielsen found in 1897:

$$\backslash gamma\; =\; 1\; -\; \backslash sum\_^\backslash infty\; (-1)^k\backslash frac.$$

In 1910, Vacca found the closely related series

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

In 1926 he found a second series:

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

$$\backslash gamma\; =\; \backslash log\backslash pi\; -\; 4\backslash log\backslash left(\backslash Gamma(\backslash tfrac34)\backslash right)\; +\; \backslash frac\; 4\; \backslash pi\; \backslash sum\_^\backslash infty\; (-1)^\backslash frac.$$

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

$$\backslash gamma\; =\; \backslash sum\_^\backslash infty\; \backslash frac$$

= \frac12 + \frac1 + \frac1 + \frac + \frac3 + \cdots,where are Gregory coefficients. This series is the special case of the expansions

convergent for

A similar series with the Cauchy numbers of the second kind is

$$\backslash gamma\; =\; 1\; -\; \backslash sum\_^\backslash infty\; \backslash frac\; =1-\; \backslash frac\; -\backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash ldots$$

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

$$\backslash gamma=\backslash sum\_^\backslash infty\backslash frac\backslash Big\backslash ,\backslash quad\; a>-1$$

where are the Bernoulli polynomials of the second kind, which are defined by the generating function

For any rational this series contains rational terms only. For example, at, it becomes

$$\backslash gamma=\backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash ldots$$Other series with the same polynomials include these examples:

$$\backslash gamma=\; -\backslash log(a+1)\; -\; \backslash sum\_^\backslash infty\backslash frac,\backslash qquad\; \backslash Re(a)>-1$$

and

$$\backslash gamma=\; -\backslash frac\; \backslash left\backslash ,\backslash qquad\; \backslash Re(a)>-1$$

where is the gamma function.

A series related to the Akiyama–Tanigawa algorithm is

$$\backslash gamma=\; \backslash log(2\backslash pi)\; -\; 2\; -\; 2\; \backslash sum\_^\backslash infty\backslash frac=\backslash log(2\backslash pi)\; -\; 2\; +\; \backslash frac\; +\; \backslash frac+\; \backslash frac\; +\; \backslash frac+\; \backslash frac\; +\; \backslash ldots$$

where are the Gregory coefficients of the second order. As a series of prime numbers:

$$\backslash gamma\; =\; \backslash lim\_\backslash left(\backslash log\; n\; -\; \backslash sum\_\backslash frac\backslash right).$$

Asymptotic expansions

equals the following asymptotic formulas (where is the th harmonic number):

• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash log\; n\; -\; \backslash frac1\; +\; \backslash frac1\; -\; \backslash frac1\; +\; \backslash cdots$ (Euler)
• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash log\backslash left(\backslash right)$ (Negoi)
• $\backslash gamma\; \backslash sim\; H\_n\; -\; \backslash frac\; -\; \backslash frac1\; +\; \backslash frac1\; -\; \backslash cdots$ (Cesàro)

The third formula is also called the Ramanujan expansion.

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

Exponential

The constant is important in number theory. It equals the following limit, where is the th prime number:

$$e^\backslash gamma\; =\; \backslash lim\_\backslash frac1\; \backslash prod\_^n\; \backslash frac.$$

This restates the third of Mertens' theorems. The numerical value of is:

Other infinite products relating to include:

These products result from the Barnes -function.

In addition,

$$e^\; =\; \backslash sqrt\; \backslash cdot\; \backslash sqrt[3]\; \backslash cdot\; \backslash sqrt[4]\; \backslash cdot\; \backslash sqrt[5]\; \backslash cdots$$

where the th factor is the th root of

$$\backslash prod\_^n\; (k+1)^.$$

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

It also holds that

$$\backslash frac=\backslash prod\_^\backslash infty\backslash left(e^\backslash left(1+\backslash frac+\backslash frac\backslash right)\backslash right).$$

Continued fraction

The continued fraction expansion of begins which has no apparent pattern. The continued fraction is known to have at least 475,006 terms, and it has infinitely many terms if and only if is irrational.

Generalizations

Euler's generalized constants are given by

$$\backslash gamma\_\backslash alpha\; =\; \backslash lim\_\backslash left(\backslash sum\_^n\; \backslash frac1\; -\; \backslash int\_1^n\; \backslash frac1\backslash ,dx\backslash right),$$

for, with as the special case . This can be further generalized to

$$c\_f\; =\; \backslash lim\_\backslash left(\backslash sum\_^n\; f(k)\; -\; \backslash int\_1^n\; f(x)\backslash ,dx\backslash right)$$

for some arbitrary decreasing function . For example,

$$f\_n(x)\; =\; \backslash frac$$

gives rise to the Stieltjes constants, and

$$f\_a(x)\; =\; x^$$

gives

$$\backslash gamma\_\; =\; \backslash frac$$

where again the limit

$$\backslash gamma\; =\; \backslash lim\_\backslash left(\backslash zeta(a)\; -\; \backslash frac1\backslash right)$$

appears.

A two-dimensional limit generalization is the Masser–Gramain constant.

Euler–Lehmer constants are given by summation of inverses of numbers in a commonmodulo class:

$$\backslash gamma(a,q)\; =\; \backslash lim\_\backslash left\; (\backslash sum\_\; \backslash frac1-\backslash frac\backslash right).$$

The basic properties are

and if the greatest common divisor then

$$q\backslash gamma(a,q)\; =\; \backslash frac\backslash gamma\backslash left(\backslash frac,\backslash frac\backslash right)-\backslash log\; d.$$

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.

Decimal digits	Author	Sources
1734	5
1735	15	Leonhard Euler
1781	16	Leonhard Euler
1790	32	Lorenzo Mascheroni, with 20–22 and 31–32 wrong
1809	22
1811	22
1812	40
1857	34	Christian Fredrik Lindman
1861	41	Ludwig Oettinger
1867	49
1871	99
1871	101	William Shanks
1877	262
1952	328
1961
1962
1962		Dura W. Sweeney
1973
1977
1980
1993
1999		Patrick Demichel and Xavier Gourdon
March 13, 2009		Alexander J. Yee & Raymond Chan
December 22, 2013		Alexander J. Yee
March 15, 2016		Peter Trueb
May 18, 2016		Ron Watkins
August 23, 2017		Ron Watkins
May 26, 2020		Seungmin Kim & Ian Cutress
May 13, 2023		Jordan Ranous & Kevin O'Brien
September 7, 2023		Andrew Sun

References

• Bretschneider . Carl Anton . 1837 . 1835 . Theoriae logarithmi integralis lineamenta nova . Crelle's Journal . 17 . 257–285 . la .
• Book: Havil, Julian . 2003 . Gamma: Exploring Euler's Constant . Princeton University Press . 978-0-691-09983-5.
• Lagarias . Jeffrey C. . 2013 . Euler's constant: Euler's work and modern developments . . 50 . 4 . 556 . 10.1090/s0273-0979-2013-01423-x . 1303.1856 . 119612431.

Footnotes

Further reading

• Borwein, Jonathan M. . David M. Bradley . Richard E. Crandall . Computational Strategies for the Riemann Zeta Function . Journal of Computational and Applied Mathematics . 2000 . 121 . 1–2 . 11 . 10.1016/s0377-0427(00)00336-8 . 2000JCoAM.121..247B . none . free . Derives as sums over Riemann zeta functions.
• Book: Finch, Steven R. . 2003 . Mathematical Constants. Encyclopedia of Mathematics and its Applications. 94. Cambridge University Press. Cambridge. 0-521-81805-2. none.
• I. . Gerst . Some series for Euler's constant . 1969 . Amer. Math. Monthly . 10.2307/2316370 . 76 . 3 . 237–275 . 2316370 . none.
• Glaisher . James Whitbread Lee . James Whitbread Lee Glaisher . 1872 . On the history of Euler's constant . Messenger of Mathematics . 1 . 25–30 . 03.0130.01 . none.
• Web site: Gourdon. Xavier. Sebah. P. . 2002 . Collection of formulae for Euler's constant, . none.
• Web site: Gourdon. Xavier. Sebah. P. . 2004 . The Euler constant: . none.
• E. A. . Karatsuba . Fast evaluation of transcendental functions . Probl. Inf. Transm. . 27 . 44 . 339–360 . 1991 . none.
• Karatsuba . E.A. . 2000 . On the computation of the Euler constant . Journal of Numerical Algorithms . 24 . 1–2 . 83–97 . 10.1023/A:1019137125281 . 21545868 . none.
• Book: Knuth, Donald . Donald Knuth . 1997 . . 3rd . Addison-Wesley . 0-201-89683-4 . 75, 107, 114, 619–620 . none.
• D. H. . Lehmer . 1975 . Euler constants for arithmetical progressions . Acta Arith. . 27 . 1 . 125–142 . 10.4064/aa-27-1-125-142 . none. free .
• Lerch . M. . 1897 . Expressions nouvelles de la constante d'Euler . Sitzungsberichte der Königlich Böhmischen Gesellschaft der Wissenschaften . 42 . 5 . none.
• Book: Mascheroni, Lorenzo . Lorenzo Mascheroni . 1790 . Adnotationes ad calculum integralem Euleri, in quibus nonnulla problemata ab Eulero proposita resolvuntur . Galeati, Ticini . none.
• Sondow . Jonathan . 2002 . A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant . math.NT/0211075 . Mathematica Slovaca . 59 . 307–314 . 10.2478/s12175-009-0127-2 . 2002math.....11075S . 16340929 . none. with an Appendix by Sergey Zlobin

External links

• • • Jonathan Sondow.
• Fast Algorithms and the FEE Method, E.A. Karatsuba (2005)
• Further formulae which make use of the constant: Gourdon and Sebah (2004).

Notes and References

1. Wolf. Marek. 6+infinity new expressions for the Euler-Mascheroni constant. 2019. 1904.09855. math.NT. "The above sum is real and convergent when zeros \rho and complex conjugate \bar{\rho} 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.
2. Boya. L.J.. Another relation between π, e, γ and ζ(n). Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. 102. 199–202. 2008. 2. 10.1007/BF03191819. "γ/2 in (10) reflects the residual (finite part) of ζ(1)/2, of course.". See formulas 1 and 10.
3. Sondow. Jonathan. Double Integrals for Euler's Constant and style (4)/(\pi) and an Analog of Hadjicostas's Formula. The American Mathematical Monthly. 112. 1. 2005. 61–65. 10.2307/30037385. 30037385. 2024-04-27.
4. Chen. Chao-Ping. Ramanujan's formula for the harmonic number. Applied Mathematics and Computation. 317. 2018. 121–128. 0096-3003. 10.1016/j.amc.2017.08.053. 2024-04-27.
5. Lodge . A. . An approximate expression for the value of 1 + 1/2 + 1/3 + ... + 1/r . Messenger of Mathematics . 30 . 1904 . 103–107 .
6. Villarino. Mark B.. Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number. 2007. 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.
7. Mortici. Cristinel. 2010. On the Stirling expansion into negative powers of a triangular number. Math. Commun.. 15. 359–364.
8. Cesàro . E. . Sur la série harmonique . Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale . 4 . 295–296 . 1885 . fr . Carilian-Goeury et Vor Dalmont.
9. Book: Bromwich , Thomas John I'Anson . An Introduction to the Theory of Infinite Series . American Mathematical Society . 2005 . 1908 . 3rd . United Kingdom . 460. See exercise 18.
10. Book: Whittaker . E. . Watson . G. . A Course of Modern Analysis . 5th . 1902 . 2021 . 271, 275 . 9781316518939 . 10.1017/9781009004091. See Examples 12.21 and 12.50 for exercises on the derivation of the integral form style \int ^{0} _{-1} ln \Gamma(z +1 )dz of the series style \sum ^{n} _{k =1} (\zeta(k))/(110 _{k}) = ln (\sqrt{ 2 \pi}) .
11. Nelsen . R. B. . Proof without Words: Sum of Reciprocals of Triangular Numbers . Mathematics Magazine . 64 . 3 . 1991 . 167. 10.1080/0025570X.1991.11977600 .
12. Book: Edwards, H. M. . Riemann's Zeta Function . Academic Press . 1974 . Pure and Applied Mathematics, Vol. 58 . 67, 159.