Euler's constant

Not to be confused withEuler's number,e ≈2.71828,the base of the natural logarithm.

Euler's constant
γ 0.57721...
General information
Type	Unknown
Fields	Number theory Analysis
History
Discovered	1734
By	Leonhard Euler
First mention	De Progressionibus harmonicis observationes
Named after	Leonhard Euler Lorenzo Mascheroni

The area of the blue region converges to Euler's constant.

Euler's constant(sometimes called theEuler–Mascheroni constant) is amathematical constant,usually denoted by the lowercase Greek lettergamma(γ), defined as thelimitingdifference between theharmonic seriesand thenatural logarithm,denoted here bylog:

$${\begin{aligned}\gamma &=\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)\\[5px]&=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,\mathrm {d} x.\end{aligned}}$$

Here,⌊·⌋represents thefloor function.

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

0.57721566490153286060651209008240243104215933593992... 

History

The constant first appeared in a 1734 paper by theSwissmathematicianLeonhard Euler,titledDe Progressionibus harmonicis observationes(Eneström Index 43). Euler used the notationsCandOfor the constant. In 1790, theItalianmathematicianLorenzo Mascheroniused the notationsAandafor the constant. The notationγappears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time, perhaps because of the constant's connection to thegamma function.For example, theGermanmathematicianCarl Anton Bretschneiderused the notationγin 1835,andAugustus De Morganused 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 theexponential integral*
• TheLaplace transform* of thenatural logarithm
• The first term of theLaurent seriesexpansion for theRiemann zeta function*, where it is the first of theStieltjes constants*
• Calculations of thedigamma function
• A product formula for thegamma function
• The asymptotic expansion of thegamma functionfor small arguments.
• An inequality forEuler's totient function
• The growth rate of thedivisor function
• Indimensional regularizationofFeynman diagramsinquantum field theory
• The calculation of theMeissel–Mertens constant
• The third ofMertens' theorems*
• Solution of the second kind toBessel's equation
• In the regularization/renormalizationof theharmonic seriesas a finite value
• Themeanof theGumbel distribution
• Theinformation entropyof theWeibullandLévydistributions, and, implicitly, of thechi-squared distributionfor one or two degrees of freedom.
• The answer to thecoupon collector's problem*
• In some formulations ofZipf's law
• A definition of thecosine integral*
• Lower bounds to aprime gap
• An upper bound onShannon entropyinquantum information theory
• Fisher–Orr modelfor genetics of adaptation in evolutionary biology
• Bardeen–Cooper–Schrieffer theory of superconductivity (BCS theory), where it appears as prefactor$2e^{\gamma }/\pi =1.134$in the BCS equation on the critical temperature.

Properties

The numberγhas not been provedalgebraicortranscendental.In fact, it is not even known whetherγisirrational.Using acontinued fractionanalysis, Papanikolaou showed in 1997 that ifγisrational,its denominator must be greater than 10²⁴⁴⁶⁶³.The ubiquity ofγrevealed by the large number of equations below makes the irrationality ofγa major open question in mathematics.

Unsolved problem in mathematics:

Is Euler's constant irrational? If so, is it transcendental?

(more unsolved problems in mathematics)

However, some progress has been made. Kurt Mahler showed in 1968 that the number${\frac {\pi }{2}}{\frac {Y_{0}(2)}{J_{0}(2)}}-\gamma$is transcendental (here,$J_{\alpha }(x)$and$Y_{\alpha }(x)$areBessel functions).In 2009 Alexander Aptekarev proved that at least one of Euler's constantγand theEuler–Gompertz constantδis irrational;Tanguy Rivoal proved in 2012 that at least one of them is transcendental.It is known that the transcendence degree of the field$\mathbb {Q} (e,\gamma,\delta )$is at least two.In 2010M. Ram Murtyand N. Saradha showed that at most one of the numbers of the form

$$\gamma (a,q)=\lim _{n\rightarrow \infty }\left(\left(\sum _{k=0}^{n}{\frac {1}{a+kq}}\right)-{\frac {\log {(a+nq})}{q}}\right)$$

withq≥ 2and1 ≤a<qis algebraic; this family includes the special caseγ(2,4) =⁠γ/4⁠.In 2013 M. Ram Murty and A. Zaytseva found a different family containingγ,which is based on sums of reciprocals of integers not divisible by a fixed list of primes, with the same property.

Euler's constant is conjectured not to be analgebraic period.

Formulas and identities

Relation to gamma function

γis related to thedigamma functionΨ,and hence thederivativeof thegamma functionΓ,when both functions are evaluated at 1. Thus:

$$-\gamma =\Gamma '(1)=\Psi (1).$$

This is equal to the limits:

$${\begin{aligned}-\gamma &=\lim _{z\to 0}\left(\Gamma (z)-{\frac {1}{z}}\right)\\&=\lim _{z\to 0}\left(\Psi (z)+{\frac {1}{z}}\right).\end{aligned}}$$

Further limit results are:

$${\begin{aligned}\lim _{z\to 0}{\frac {1}{z}}\left({\frac {1}{\Gamma (1+z)}}-{\frac {1}{\Gamma (1-z)}}\right)&=2\gamma \\\lim _{z\to 0}{\frac {1}{z}}\left({\frac {1}{\Psi (1-z)}}-{\frac {1}{\Psi (1+z)}}\right)&={\frac {\pi ^{2}}{3\gamma ^{2}}}.\end{aligned}}$$

A limit related to thebeta function(expressed in terms ofgamma functions) is

$${\begin{aligned}\gamma &=\lim _{n\to \infty }\left({\frac {\Gamma \left({\frac {1}{n}}\right)\Gamma (n+1)\,n^{1+{\frac {1}{n}}}}{\Gamma \left(2+n+{\frac {1}{n}}\right)}}-{\frac {n^{2}}{n+1}}\right)\\&=\lim \limits _{m\to \infty }\sum _{k=1}^{m}{m \choose k}{\frac {(-1)^{k}}{k}}\log {\big (}\Gamma (k+1){\big )}.\end{aligned}}$$

Relation to the zeta function

γcan also be expressed as aninfinite sumwhose terms involve theRiemann zeta functionevaluated at positive integers:

$${\begin{aligned}\gamma &=\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{m}}\\&=\log {\frac {4}{\pi }}+\sum _{m=2}^{\infty }(-1)^{m}{\frac {\zeta (m)}{2^{m-1}m}}.\end{aligned}}$$ The constant$\gamma$can also be expressed in terms of the sum of the reciprocals ofnon-trivial zeros$\rho$of the zeta function:

$\gamma =\log 4\pi +\sum _{\rho }{\frac {2}{\rho }}-2$

Other series related to the zeta function include:

$${\begin{aligned}\gamma &={\tfrac {3}{2}}-\log 2-\sum _{m=2}^{\infty }(-1)^{m}\,{\frac {m-1}{m}}{\big (}\zeta (m)-1{\big )}\\&=\lim _{n\to \infty }\left({\frac {2n-1}{2n}}-\log n+\sum _{k=2}^{n}\left({\frac {1}{k}}-{\frac {\zeta (1-k)}{n^{k}}}\right)\right)\\&=\lim _{n\to \infty }\left({\frac {2^{n}}{e^{2^{n}}}}\sum _{m=0}^{\infty }{\frac {2^{mn}}{(m+1)!}}\sum _{t=0}^{m}{\frac {1}{t+1}}-n\log 2+O\left({\frac {1}{2^{n}\,e^{2^{n}}}}\right)\right).\end{aligned}}$$

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

$${\begin{aligned}\gamma &=\lim _{s\to 1^{+}}\sum _{n=1}^{\infty }\left({\frac {1}{n^{s}}}-{\frac {1}{s^{n}}}\right)\\&=\lim _{s\to 1}\left(\zeta (s)-{\frac {1}{s-1}}\right)\\&=\lim _{s\to 0}{\frac {\zeta (1+s)+\zeta (1-s)}{2}}\end{aligned}}$$

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

$$\gamma =\lim _{n\to \infty }{\frac {1}{n}}\,\sum _{k=1}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right)$$

where⌈ ⌉areceilingbrackets. This formula indicates that when taking any positive integernand dividing it by each positive integerkless thann,the average fraction by which the quotientn/kfalls short of the next integer tends toγ(rather than 0.5) asntends to infinity.

Closely related to this is therational zeta seriesexpression. By taking separately the first few terms of the series above, one obtains an estimate for the classical series limit:

$$\gamma =\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n-\sum _{m=2}^{\infty }{\frac {\zeta (m,n+1)}{m}}\right),$$

whereζ(s,k)is theHurwitz zeta function.The sum in this equation involves theharmonic numbers,H_n.Expanding some of the terms in the Hurwitz zeta function gives:

$$H_{n}=\log(n)+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\varepsilon,$$ where0 <ε<⁠1/252n⁶⁠.

γcan also be expressed as follows whereAis theGlaisher–Kinkelin constant:

$$\gamma =12\,\log(A)-\log(2\pi )+{\frac {6}{\pi ^{2}}}\,\zeta '(2)$$

γcan also be expressed as follows, which can be proven by expressing thezeta functionas aLaurent series:

$$\gamma =\lim _{n\to \infty }\left(-n+\zeta \left({\frac {n+1}{n}}\right)\right)$$

Relation to triangular numbers

Numerous formulations have been derived that express$\gamma$in terms of sums and logarithms oftriangular numbers.One of the earliest of these is a formulafor the$n$thharmonic numberattributed toSrinivasa Ramanujanwhere$\gamma$is related to$\textstyle \ln 2T_{k}$in a series that considers the powers of$\textstyle {\frac {1}{T_{k}}}$(an earlier, less-generalizable proofbyErnesto Cesàrogives the first two terms of the series, with an error term):

${\begin{aligned}\gamma &=H_{u}-{\frac {1}{2}}\ln 2T_{u}-\sum _{k=1}^{v}{\frac {R(k)}{T_{u}^{k}}}-\Theta _{v}\,{\frac {R(v+1)}{T_{u}^{v+1}}}\end{aligned}}$

FromStirling's approximationfollows a similar series:

$\gamma =\ln 2\pi -\sum _{k=2}^{n}{\frac {\zeta (k)}{T_{k}}}$

The series of inverse triangular numbers also features in the study of theBasel problemposed byPietro Mengoli.Mengoli proved that$\textstyle \sum _{k=1}^{\infty }{\frac {1}{2T_{k}}}=1$,a resultJacob Bernoullilater used to estimate thevalueof$\zeta (2)$,placing it between$1$and$\textstyle \sum _{k=1}^{\infty }{\frac {2}{2T_{k}}}=\sum _{k=1}^{\infty }{\frac {1}{T_{k}}}=2$.This identity appears in a formula used byBernhard Riemannto computeroots of the zeta function,where$\gamma$is expressed in terms of the sum of roots$\rho$plus the difference between Boya's expansion and the series of exactunit fractions$\textstyle \sum _{k=1}^{n}{\frac {1}{T_{k}}}$:

$\gamma -\ln 2=\ln 2\pi +\sum _{\rho }{\frac {2}{\rho }}-\sum _{k=1}^{n}{\frac {1}{T_{k}}}$

Integrals

γequals the value of a number of definiteintegrals:

$${\begin{aligned}\gamma &=-\int _{0}^{\infty }e^{-x}\log x\,dx\\&=-\int _{0}^{1}\log \left(\log {\frac {1}{x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{x\cdot e^{x}}}\right)dx\\&=\int _{0}^{1}{\frac {1-e^{-x}}{x}}\,dx-\int _{1}^{\infty }{\frac {e^{-x}}{x}}\,dx\\&=\int _{0}^{1}\left({\frac {1}{\log x}}+{\frac {1}{1-x}}\right)dx\\&=\int _{0}^{\infty }\left({\frac {1}{1+x^{k}}}-e^{-x}\right){\frac {dx}{x}},\quad k>0\\&=2\int _{0}^{\infty }{\frac {e^{-x^{2}}-e^{-x}}{x}}\,dx,\\&=\log {\frac {\pi }{4}}-\int _{0}^{\infty }{\frac {\log x}{\cosh ^{2}x}}\,dx,\\&=\int _{0}^{1}H_{x}\,dx,\\&={\frac {1}{2}}+\int _{0}^{\infty }\log \left(1+{\frac {\log \left(1+{\frac {1}{t}}\right)^{2}}{4\pi ^{2}}}\right)dt\\&=1-\int _{0}^{1}\{1/x\}dx\end{aligned}}$$ whereH_xis thefractional harmonic number,and$\{1/x\}$is thefractional partof$1/x$.

The third formula in the integral list can be proved in the following way:

$${\begin{aligned}&\int _{0}^{\infty }\left({\frac {1}{e^{x}-1}}-{\frac {1}{xe^{x}}}\right)dx=\int _{0}^{\infty }{\frac {e^{-x}+x-1}{x[e^{x}-1]}}dx=\int _{0}^{\infty }{\frac {1}{x[e^{x}-1]}}\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m+1}}{(m+1)!}}dx\\[2pt]&=\int _{0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![e^{x}-1]}}dx=\sum _{m=1}^{\infty }\int _{0}^{\infty }{\frac {(-1)^{m+1}x^{m}}{(m+1)![e^{x}-1]}}dx=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}\int _{0}^{\infty }{\frac {x^{m}}{e^{x}-1}}dx\\[2pt]&=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{(m+1)!}}m!\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\zeta (m+1)=\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}\sum _{n=1}^{\infty }{\frac {1}{n^{m+1}}}=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}\\[2pt]&=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{m+1}}{m+1}}{\frac {1}{n^{m+1}}}=\sum _{n=1}^{\infty }\left[{\frac {1}{n}}-\log \left(1+{\frac {1}{n}}\right)\right]=\gamma \end{aligned}}$$

The integral on the second line of the equation stands for theDebye functionvalue of+∞,which ism!ζ(m+ 1).

Definite integrals in whichγappears include:

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

One can expressγusing a special case ofHadjicostas's formulaas adouble integralwith equivalent series:

$${\begin{aligned}\gamma &=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1-xy)\log xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right).\end{aligned}}$$

An interesting comparison by Sondowis the double integral and alternating series

$${\begin{aligned}\log {\frac {4}{\pi }}&=\int _{0}^{1}\int _{0}^{1}{\frac {x-1}{(1+xy)\log xy}}\,dx\,dy\\&=\sum _{n=1}^{\infty }\left((-1)^{n-1}\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right)\right).\end{aligned}}$$

It shows thatlog⁠4/π⁠may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series

$${\begin{aligned}\gamma &=\sum _{n=1}^{\infty }{\frac {N_{1}(n)+N_{0}(n)}{2n(2n+1)}}\\\log {\frac {4}{\pi }}&=\sum _{n=1}^{\infty }{\frac {N_{1}(n)-N_{0}(n)}{2n(2n+1)}},\end{aligned}}$$

whereN₁(n)andN₀(n)are the number of 1s and 0s, respectively, in thebase 2expansion ofn.

We also haveCatalan's 1875 integral

$$\gamma =\int _{0}^{1}\left({\frac {1}{1+x}}\sum _{n=1}^{\infty }x^{2^{n}-1}\right)\,dx.$$

Series expansions

In general,

$$\gamma =\lim _{n\to \infty }\left({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\ldots +{\frac {1}{n}}-\log(n+\alpha )\right)\equiv \lim _{n\to \infty }\gamma _{n}(\alpha )$$

for anyα> −n.However, the rate of convergence of this expansion depends significantly onα.In particular,γ_n(1/2)exhibits much more rapid convergence than the conventional expansionγ_n(0).This is because

$${\frac {1}{2(n+1)}}<\gamma _{n}(0)-\gamma <{\frac {1}{2n}},$$

while

$${\frac {1}{24(n+1)^{2}}}<\gamma _{n}(1/2)-\gamma <{\frac {1}{24n^{2}}}.$$

Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.

Euler showed that the followinginfinite seriesapproachesγ: $$\gamma =\sum _{k=1}^{\infty }\left({\frac {1}{k}}-\log \left(1+{\frac {1}{k}}\right)\right).$$

The series forγis equivalent to a seriesNielsenfound in 1897:

$$\gamma =1-\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k+1}}.$$

In 1910,Vaccafound the closely related series

$${\begin{aligned}\gamma &=\sum _{k=1}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}\\[5pt]&={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-{\tfrac {1}{9}}+{\tfrac {1}{10}}-{\tfrac {1}{11}}+\cdots -{\tfrac {1}{15}}\right)+\cdots,\end{aligned}}$$

wherelog₂is thelogarithm to base 2and⌊⌋is thefloor function.

This can be generalized to:

$$\gamma =\sum _{k=1}^{\infty }{\frac {\left\lfloor \log _{B}k\right\rfloor }{k}}\varepsilon (k)$$where:$$\varepsilon (k)={\begin{cases}B-1,&{\text{if }}B\mid n\\-1,&{\text{if }}B\nmid n\end{cases}}$$

In 1926 Vacca found a second series:

$${\begin{aligned}\gamma +\zeta (2)&=\sum _{k=2}^{\infty }\left({\frac {1}{\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}}-{\frac {1}{k}}\right)\\[5pt]&=\sum _{k=2}^{\infty }{\frac {k-\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}{k\left\lfloor {\sqrt {k}}\right\rfloor ^{2}}}\\[5pt]&={\frac {1}{2}}+{\frac {2}{3}}+{\frac {1}{2^{2}}}\sum _{k=1}^{2\cdot 2}{\frac {k}{k+2^{2}}}+{\frac {1}{3^{2}}}\sum _{k=1}^{3\cdot 2}{\frac {k}{k+3^{2}}}+\cdots \end{aligned}}$$

From theMalmsten–Kummerexpansion for the logarithm of the gamma functionwe get:

$$\gamma =\log \pi -4\log \left(\Gamma ({\tfrac {3}{4}})\right)+{\frac {4}{\pi }}\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {\log(2k+1)}{2k+1}}.$$

Ramanujan, in hislost notebookgave a series that approachesγ:

$$\gamma =\log 2-\sum _{n=1}^{\infty }\sum _{k={\frac {3^{n-1}+1}{2}}}^{\frac {3^{n}-1}{2}}{\frac {2n}{(3k)^{3}-3k}}$$

An important expansion for Euler's constant is due toFontanaandMascheroni

$$\gamma =\sum _{n=1}^{\infty }{\frac {|G_{n}|}{n}}={\frac {1}{2}}+{\frac {1}{24}}+{\frac {1}{72}}+{\frac {19}{2880}}+{\frac {3}{800}}+\cdots,$$ whereG_nareGregory coefficients.This series is the special casek= 1of the expansions

$${\begin{aligned}\gamma &=H_{k-1}-\log k+\sum _{n=1}^{\infty }{\frac {(n-1)!|G_{n}|}{k(k+1)\cdots (k+n-1)}}&&\\&=H_{k-1}-\log k+{\frac {1}{2k}}+{\frac {1}{12k(k+1)}}+{\frac {1}{12k(k+1)(k+2)}}+{\frac {19}{120k(k+1)(k+2)(k+3)}}+\cdots &&\end{aligned}}$$

convergent fork= 1, 2,...

A similar series with the Cauchy numbers of the second kindC_nis

$$\gamma =1-\sum _{n=1}^{\infty }{\frac {C_{n}}{n\,(n+1)!}}=1-{\frac {1}{4}}-{\frac {5}{72}}-{\frac {1}{32}}-{\frac {251}{14400}}-{\frac {19}{1728}}-\ldots$$

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

$$\gamma =\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2n}}{\Big \{}\psi _{n}(a)+\psi _{n}{\Big (}-{\frac {a}{1+a}}{\Big )}{\Big \}},\quad a>-1$$

whereψ_n(a)are theBernoulli polynomials of the second kind,which are defined by the generating function

$${\frac {z(1+z)^{s}}{\log(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(s),\qquad |z|<1.$$

For any rationalathis series contains rational terms only. For example, ata= 1,it becomes

$$\gamma ={\frac {3}{4}}-{\frac {11}{96}}-{\frac {1}{72}}-{\frac {311}{46080}}-{\frac {5}{1152}}-{\frac {7291}{2322432}}-{\frac {243}{100352}}-\ldots$$ Other series with the same polynomials include these examples:

$$\gamma =-\log(a+1)-\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)}{n}},\qquad \Re (a)>-1$$

and

$$\gamma =-{\frac {2}{1+2a}}\left\{\log \Gamma (a+1)-{\frac {1}{2}}\log(2\pi )+{\frac {1}{2}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n+1}(a)}{n}}\right\},\qquad \Re (a)>-1$$

whereΓ(a)is thegamma function.

A series related to the Akiyama–Tanigawa algorithm is

$$\gamma =\log(2\pi )-2-2\sum _{n=1}^{\infty }{\frac {(-1)^{n}G_{n}(2)}{n}}=\log(2\pi )-2+{\frac {2}{3}}+{\frac {1}{24}}+{\frac {7}{540}}+{\frac {17}{2880}}+{\frac {41}{12600}}+\ldots$$

whereG_n(2)are theGregory coefficientsof the second order.

As a series ofprime numbers:

$$\gamma =\lim _{n\to \infty }\left(\log n-\sum _{p\leq n}{\frac {\log p}{p-1}}\right).$$

Asymptotic expansions

γequals the following asymptotic formulas (whereH_nis thenthharmonic number):

• ${\textstyle \gamma \sim H_{n}-\log n-{\frac {1}{2n}}+{\frac {1}{12n^{2}}}-{\frac {1}{120n^{4}}}+\cdots }$(Euler)
• ${\textstyle \gamma \sim H_{n}-\log \left({n+{\frac {1}{2}}+{\frac {1}{24n}}-{\frac {1}{48n^{2}}}+\cdots }\right)}$(Negoi)
• ${\textstyle \gamma \sim H_{n}-{\frac {\log n+\log(n+1)}{2}}-{\frac {1}{6n(n+1)}}+{\frac {1}{30n^{2}(n+1)^{2}}}-\cdots }$(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):

$${\begin{aligned}\sum _{n=1}^{\infty }\log n+\gamma -H_{n}+{\frac {1}{2n}}&={\frac {\log(2\pi )-1-\gamma }{2}}\\\sum _{n=1}^{\infty }\log {\sqrt {n(n+1)}}+\gamma -H_{n}&={\frac {\log(2\pi )-1}{2}}-\gamma \\\sum _{n=1}^{\infty }(-1)^{n}{\Big (}\log n+\gamma -H_{n}{\Big )}&={\frac {\log \pi -\gamma }{2}}\end{aligned}}$$

Exponential

The constante^γis important in number theory. Its numerical value is:

1.78107241799019798523650410310717954916964521430343....

e^γequals the followinglimit,wherep_nis thenthprime number:

$$e^{\gamma }=\lim _{n\to \infty }{\frac {1}{\log p_{n}}}\prod _{i=1}^{n}{\frac {p_{i}}{p_{i}-1}}.$$

This restates the third ofMertens' theorems.

We further have the following product involving the three constantse,πandγ:

$${\frac {\pi ^{2}}{6e^{\gamma }}}=\lim _{n\to \infty }{\frac {1}{\log p_{n}}}\prod _{i=1}^{n}{\frac {p_{i}}{p_{i}+1}}.$$

Otherinfinite productsrelating toe^γinclude:

$${\begin{aligned}{\frac {e^{1+{\frac {\gamma }{2}}}}{\sqrt {2\pi }}}&=\prod _{n=1}^{\infty }e^{-1+{\frac {1}{2n}}}\left(1+{\frac {1}{n}}\right)^{n}\\{\frac {e^{3+2\gamma }}{2\pi }}&=\prod _{n=1}^{\infty }e^{-2+{\frac {2}{n}}}\left(1+{\frac {2}{n}}\right)^{n}.\end{aligned}}$$

These products result from theBarnesG-function.

In addition,

$$e^{\gamma }={\sqrt {\frac {2}{1}}}\cdot {\sqrt[{3}]{\frac {2^{2}}{1\cdot 3}}}\cdot {\sqrt[{4}]{\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}}\cdot {\sqrt[{5}]{\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}}\cdots$$

where thenth factor is the(n+ 1)th root of

$$\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}.$$

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

It also holds that

$${\frac {e^{\frac {\pi }{2}}+e^{-{\frac {\pi }{2}}}}{\pi e^{\gamma }}}=\prod _{n=1}^{\infty }\left(e^{-{\frac {1}{n}}}\left(1+{\frac {1}{n}}+{\frac {1}{2n^{2}}}\right)\right).$$

Ife^γis a rational number, then its denominator must be greater than 10¹⁵⁰⁰⁰.

Continued fraction

The simplecontinued fractionexpansion of Euler's constant is given by:

$\gamma =0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{4+\dots }}}}}}}}}}}}}}$

which has noapparentpattern. It is known to have at least 4,800,000,000 terms,and it has infinitely many termsif and only ifγis irrational.

Numerical evidence suggests that Euler's constant is among the numbers for which thegeometric meanof the continued fraction terms converges toKhinchin's constant.Similarly, when$p_{n}/q_{n}$are the convergents of the continued fraction, the limit$\lim _{n\to \infty }q_{n}^{1/n}$appears to converge toLévy's constant.However neither of these limits has been proven.

Generalizations

Stieltjes constants

Main article:Stieltjes constants

Euler's generalized constantsabm(-$\alpha$)forα> 0.

Euler's generalized constantsare given by

$$\gamma _{\alpha }=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k^{\alpha }}}-\int _{1}^{n}{\frac {1}{x^{\alpha }}}\,dx\right)$$

for0 <α< 1,withγas the special caseα= 1.Extending forα> 1gives:

$$\gamma _{\alpha }=\zeta (\alpha )-{\frac {1}{\alpha -1}}$$

with again the limit:

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

This can be further generalized to

$$c_{f}=\lim _{n\to \infty }\left(\sum _{k=1}^{n}f(k)-\int _{1}^{n}f(x)\,dx\right)$$

for some arbitrary decreasing functionf.Setting

$$f_{n}(x)={\frac {(\log x)^{n}}{x}}$$

gives rise to theStieltjes constants$\gamma _{n}$,that occur in theLaurent seriesexpansion of theRiemann zeta function:

$\zeta (1+s)={\frac {1}{s}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}s^{n}.$

with$\gamma _{0}=\gamma =0.577\dots$

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

Euler–Lehmer constantsare given by summation of inverses of numbers in a common modulo class:

$$\gamma (a,q)=\lim _{x\to \infty }\left(\sum _{0<n\leq x \atop n\equiv a{\pmod {q}}}{\frac {1}{n}}-{\frac {\log x}{q}}\right).$$

The basic properties are

$${\begin{aligned}&\gamma (0,q)={\frac {\gamma -\log q}{q}},\\&\sum _{a=0}^{q-1}\gamma (a,q)=\gamma,\\&q\gamma (a,q)=\gamma -\sum _{j=1}^{q-1}e^{-{\frac {2\pi aij}{q}}}\log \left(1-e^{\frac {2\pi ij}{q}}\right),\end{aligned}}$$

and if thegreatest common divisorgcd(a,q) =dthen

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

Euler-constant function

The Euler-constant function can be defined as:$$\gamma (z)=\sum _{n=1}^{\infty }z^{n-1}\left({\frac {1}{n}}-\log {\frac {n+1}{n}}\right)$$for$|z|\leq 1$which generalizes the series for Euler's constant for$z=1$and the ''alternating constant'' for$z=-1$.It's domain can be extended via the following integral for$z\in \mathbb {C} \setminus (1,\infty )$:$$\gamma (z)=\int _{0}^{1}\int _{0}^{1}{\frac {1-x}{(xyz-1)\log xy}}\mathrm {d} x\mathrm {d} y$$This gives among others the following specific values:

$\gamma (1)=\gamma$

$\gamma ({\tfrac {1}{2}})=2\ln {\tfrac {2}{\sigma }}$

$\gamma (0)=1-\ln 2$

$\gamma (-1)=\ln {\tfrac {4}{\pi }}$

$\gamma '(-1)={\tfrac {11}{6}}\ln 2+6\ln A-{\tfrac {3}{2}}\ln \pi -1$

where$\sigma$isSomos' quadratic recurrence constantand$A$is Glaisher's constant.

Masser-Gramain constant

A two-dimensional generalization of Euler's constant is theMasser-Gramain constant.It is defined as the following limiting difference:

$\delta =\lim _{n\to \infty }\left(-\log n+\sum _{k=2}^{n}{\frac {1}{\pi r_{k}^{2}}}\right)$

where$r_{k}$is the smallest radius of a disk in the complex plane containing at least$k$Gaussian integers.

The following bounds have been established:$1.819776<\delta <1.819833$.

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...1811209008239when the correct value is...0651209008240.

Published decimal expansions ofγ

Date	Decimal digits	Author	Sources
1734	5	Leonhard Euler
1735	15	Leonhard Euler
1781	16	Leonhard Euler
1790	32	Lorenzo Mascheroni,with 20–22 and 31–32 wrong
1809	22	Johann G. von Soldner
1811	22	Carl Friedrich Gauss
1812	40	Friedrich Bernhard Gottfried Nicolai
1857	34	Christian Fredrik Lindman
1861	41	Ludwig Oettinger
1867	49	William Shanks
1871	99	James W.L. Glaisher
1871	101	William Shanks
1877	262	J. C. Adams
1952	328	John William Wrench Jr.
1961	1050	Helmut Fischer andKarl Zeller
1962	1271	Donald Knuth
1962	3566	Dura W. Sweeney
1973	4879	William A. Beyer andMichael S. Waterman
1977	20700	Richard P. Brent
1980	30100	Richard P. Brent &Edwin M. McMillan
1993	172000	Jonathan Borwein
1999	108000000	Patrick Demichel and Xavier Gourdon
March 13, 2009	29844489545	Alexander J. Yee & Raymond Chan
December 22, 2013	119377958182	Alexander J. Yee
March 15, 2016	160000000000	Peter Trueb
May 18, 2016	250000000000	Ron Watkins
August 23, 2017	477511832674	Ron Watkins
May 26, 2020	600000000100	Seungmin Kim & Ian Cutress
May 13, 2023	700000000000	Jordan Ranous & Kevin O'Brien
September 7, 2023	1337000000000	Andrew Sun

See also

• Harmonic series
• Riemann zeta function
• Stieltjes constants
• Meissel-Mertens constant

References

• 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.ISBN978-0-691-09983-5.
• 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.S2CID119612431.

Further reading

• 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.Derivesγas sums over Riemann zeta functions.
• Finch, Steven R. (2003).Mathematical Constants.Encyclopedia of Mathematics and its Applications. Vol. 94. Cambridge: Cambridge University Press.ISBN0-521-81805-2.
• Gerst, I. (1969). "Some series for Euler's constant".Amer. Math. Monthly.76(3): 237–275.doi:10.2307/2316370.JSTOR2316370.
• Glaisher, James Whitbread Lee(1872). "On the history of Euler's constant".Messenger of Mathematics.1:25–30.JFM03.0130.01.
• Gourdon, Xavier; Sebah, P. (2002)."Collection of formulae for Euler's constant,γ".
• Gourdon, Xavier; Sebah, P. (2004)."The Euler constant:γ".
• 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 constantγ".Journal of Numerical Algorithms.24(1–2): 83–97.doi:10.1023/A:1019137125281.S2CID21545868.
• Knuth, Donald(1997).The Art of Computer Programming, Vol. 1(3rd ed.). Addison-Wesley. pp. 75, 107, 114, 619–620.ISBN0-201-89683-4.
• 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.S2CID16340929.with an Appendix bySergey Zlobin

External links

• "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).