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Euler-Mascheroni Constant

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The Euler-Mascheroni constant gamma, sometimes also called 'Euler's constant' or 'the Euler constant' (but not to be confused with the constant e=2.718281...) is defined as the limit of the sequence

gamma=lim_(n->infty)(sum_(k=1)^(n)1/k-lnn)
(1)
=lim_(n->infty)(H_n-lnn),
(2)

where H_n is a harmonic number (Graham et al. 1994, p. 278). It was first defined by Euler (1735), who used the letter C and stated that it was "worthy of serious consideration" (Havil 2003, pp. xx and 51). The symbol gamma was first used by Mascheroni (1790).

gamma has the numerical value

gamma=0.577215664901532860606512090082402431042...
(3)

(OEIS A001620), and is implemented in the Wolfram Language as EulerGamma.

It is not known if this constant is irrational, let alone transcendental (Wells 1986, p. 28). The famous English mathematician G. H. Hardy is alleged to have offered to give up his Savilian Chair at Oxford to anyone who proved gamma to be irrational (Havil 2003, p. 52), although no written reference for this quote seems to be known. Hilbert mentioned the irrationality of gamma as an unsolved problem that seems "unapproachable" and in front of which mathematicians stand helpless (Havil 2003, p. 97). Conway and Guy (1996) are "prepared to bet that it is transcendental," although they do not expect a proof to be achieved within their lifetimes. If gamma is a simple fraction a/b, then it is known that b>10^(10000) (Brent 1977; Wells 1986, p. 28), which was subsequently improved by T. Papanikolaou to b>10^(242080) (Havil 2003, p. 97).

The Euler-Mascheroni constant continued fraction is given by [0, 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (OEIS A002852).

The Engel expansion of gamma is given by 2, 7, 13, 19, 85, 2601, 9602, 46268, 4812284, ... (OEIS A053977).

The Euler-Mascheroni constant arises in many integrals

gamma=-int_0^inftye^(-x)lnxdx
(4)
=-int_0^1lnln(1/x)dx
(5)
=int_0^infty(1/(1-e^(-x))-1/x)e^(-x)dx
(6)
=int_0^infty1/x(1/(1+x)-e^(-x))dx
(7)

(Whittaker and Watson 1990, p. 246). Integrals that give gamma in combination with other simple constants include

int_0^inftye^(-x^2)lnxdx=-1/4sqrt(pi)(gamma+2ln2)
(8)
int_0^inftye^(-x)(lnx)^2dx=gamma^2+1/6pi^2.
(9)

Double integrals include

gamma=int_0^1int_0^1(x-1)/((1-xy)ln(xy))dxdy
(10)

(Sondow 2003, 2005; Borwein et al. 2004, p. 49). An interesting analog of equation (10) is given by

ln(4/pi)=sum_(n=1)^(infty)(-1)^(n-1)[1/n-ln((n+1)/n)]
(11)
=int_0^1int_0^1(x-1)/((1+xy)ln(xy))dxdy
(12)
=0.241564...
(13)

(OEIS A094640; Sondow 2005).

gamma is also given by Mertens theorem

e^gamma=lim_(n->infty)1/(lnp_n)product_(i=1)^n1/(1-1/(p_i)),
(14)

where the product is over primes p. By taking the logarithm of both sides, an explicit formula for gamma is obtained,

gamma=lim_(x->infty)[sum_(p<=x)ln(1/(1-1/p))-lnlnx].
(15)

It is also given by series

gamma=sum_(k=1)^infty[1/k-ln(1+1/k)]
(16)

due to Euler, which follows from equation (1) by first replacing lnn by ln(n+1), which works since

lim_(n->infty)[ln(n+1)-lnn]=lim_(n->infty)ln(1+1/n)=0,
(17)

and then substituting the telescoping sum

sum_(k=1)^nln(1+1/k)
(18)

for ln(n+1), which is its sum since again

ln(1+1/k)=ln(k+1)-lnk,
(19)

obtaining

gamma=lim_(n->infty)[sum_(k=1)^(n)1/k-sum_(k=1)^(n)ln(1+1/k)]
(20)
=lim_(n->infty)sum_(k=1)^(n)[1/k-ln(1+1/k)]
(21)

which equals equation (◇).

Other series include

gamma=sum_(n=2)^(infty)(-1)^n(zeta(n))/n
(22)
=ln(4/pi)+sum_(n=1)^(infty)((-1)^(n-1)zeta(n+1))/(2^n(n+1))
(23)

(Gourdon and Sebah 2003, p. 3), where zeta(z) is the Riemann zeta function, and

gamma=sum_(n=1)^infty(-1)^n(|_lgn_|)/n
(24)

(Vacca 1910, Gerst 1969), where lg is the logarithm to base 2 and |_x_| is the floor function. Nielsen (1897) earlier gave a series equivalent to (24),

gamma=1-sum_(n=1)^inftysum_(k=2^(n-1))^(2^n-1)n/((2k+1)(2k+2)).
(25)

To see the equivalence of (25) with (24), expand

1/((2k+1)(2k+2))=1/(2k+1)-1/(2k+2)
(26)

and add

0=-1/2+1/4+1/8+1/(16)+...
(27)

to Nielsen's equation to get Vacca's formula.

The sums

gamma=sum_(n=1)^(infty)sum_(k=2^n)^(infty)((-1)^k)/k
(28)
=sum_(k=1)^(infty)1/(2^(k+1))sum_(j=0)^(k-1)(2^(k-j)+j; j)^(-1)
(29)

(Gosper 1972, with k-j replacing the undefined i; Bailey and Crandall 2001) can be obtained from equation (24) by rewriting as a double series, then applying Euler's series transformation to each of these series and adding to get equation (29). Here, (n; k) is a binomial coefficient, and rearranging the conditionally convergent series is permitted because the plus and minus terms can first be grouped in pairs, the resulting series of positive numbers rearranged, and then the series ungrouped back to plus and minus terms.

The double series (28) is equivalent to Catalan's integral

gamma=int_0^11/(1+x)sum_(n=1)^inftyx^(2^n-1)dx.
(30)

To see the equivalence, expand 1/(1+x) in a geometric series, multiply by x^(2^n-1), and integrate termwise (Sondow and Zudilin 2003).

Other series for gamma include

gamma=3/2-ln2-sum_(m=2)^infty(-1)^m(m-1)/m[zeta(m)-1]
(31)

(Flajolet and Vardi 1996), and

gamma=(2^n)/(e^(2^n))sum_(m=0)^infty(2^(mn))/((m+1)!)sum_(t=0)^m1/(t+1)-nln2+O(1/(2^ne^(2^n))),
(32)

(Bailey 1988), which is an improvement over Sweeney (1963).

A rapidly converging limit for gamma is given by

gamma=lim_(n->infty)[(2n-1)/(2n)-lnn+sum_(k=2)^(n)(1/k-(zeta(1-k))/(n^k))]
(33)
=lim_(n->infty)[(2n-1)/(2n)-lnn+sum_(k=2)^(n)1/k(1+(B_k)/(n^k))],
(34)

where B_k is a Bernoulli number (C. Stingley, pers. comm., July 11, 2003).

Another limit formula is given by

gamma=-lim_(n->infty)[(Gamma(1/n)Gamma(n+1)n^(1+1/n))/(Gamma(2+n+1/n))-(n^2)/(n+1)]
(35)

(P. Walker, pers. comm., Mar. 17, 2004). An even more amazing limit is given by

gamma=lim_(x->infty)zeta(zeta(z))-2^x+(4/3)^x+1
(36)

(B. Cloitre, pers. comm., Oct. 4, 2005), where zeta(z) is the Riemann zeta function.

Another connection with the primes was provided by Dirichlet's 1838 proof that the average number of divisors d(n)=sigma_0(n) of all numbers from 1 to n is asymptotic to

(sum_(k=1)^(n)d(k))/n∼lnn+2gamma-1
(37)

(Conway and Guy 1996). de la Vallée Poussin (1898) proved that, if a large number n is divided by all primes <=n, then the average amount by which the quotient is less than the next whole number is gamma.

An elegant identity for gamma is given by

gamma=(S_0(z)-K_0(z))/(I_0(z))-ln(1/2z),
(38)

where I_0(z) is a modified Bessel function of the first kind, K_0(z) is a modified Bessel function of the second kind, and

S_0(z)=sum_(k=0)^infty((1/2z)^(2k)H_k)/((k!)^2),
(39)

where H_n is a harmonic number (Borwein and Borwein 1987, p. 336; Borwein and Bailey 2003, p. 138). This gives an efficient iterative algorithm for gamma by computing

B_k=(B_(k-1)n^2)/(k^2)
(40)
A_k=1/k((A_(k-1)n^2)/k+B_k)
(41)
U_k=U_(k-1)+A_k
(42)
V_k=V_(k-1)+B_k
(43)

with A_0=-lnn, B_0=1, U_0=A_0, and V_0=1 (Borwein and Bailey 2003, pp. 138-139).

Reformulating this identity gives the limit

lim_(n->infty)[sum_(k=0)^infty(((n^k)/(k!))^2H_k)/(sum_(k=0)^(infty)((n^k)/(k!))^2)-lnn]=gamma
(44)

(Brent and McMillan 1980; Trott 2004, p. 21).

Infinite products involving gamma also arise from the Barnes G-function with positive integer n. The cases G(2) and G(3) give

product_(n=1)^(infty)e^(-1+1/(2n))(1+1/n)^n=(e^(1+gamma/2))/(sqrt(2pi))
(45)
product_(n=1)^(infty)e^(-2+2/n)(1+2/n)^n=(e^(3+2gamma))/(2pi).
(46)

The Euler-Mascheroni constant is also given by the expressions

gamma=-Gamma^'(1)
(47)
=-psi_0(1),
(48)

where psi_0(x) is the digamma function (Whittaker and Watson 1990, p. 236),

gamma=lim_(s->1)[zeta(s)-1/(s-1)]
(49)

(Whittaker and Watson 1990, p. 271), the antisymmetric limit form

gamma=lim_(s->1^+)sum_(n=1)^infty(1/(n^s)-1/(s^n))
(50)

(Sondow 1998), and

gamma=lim_(x->infty)[x-Gamma(1/x)]
(51)

(Le Lionnais 1983).

The difference between the nth convergent in equation (◇) and gamma is given by

sum_(k=1)^n1/k-lnn-gamma=int_n^infty(x-|_x_|)/(x^2)dx,
(52)

where |_x_| is the floor function, and satisfies the inequality

1/(2(n+1))<sum_(k=1)^n1/k-lnn-gamma<1/(2n)
(53)

(Young 1991).

The symbol gamma is sometimes also used for

gamma^'=e^gamma approx 1.781072
(54)

(OEIS A073004; Gradshteyn and Ryzhik 2000, p. xxvii).

There is a the curious radical representation

e^gamma=(2/1)^(1/2)((2^2)/(1·3))^(1/3)((2^3·4)/(1·3^3))^(1/4)((2^4·4^4)/(1·3^6·5))^(1/5)...,
(55)

which is related to the double series

gamma=sum_(n=1)^infty1/nsum_(k=0)^(n-1)(-1)^(k+1)(n-1; k)ln(k+1)
(56)

and (n; k) a binomial coefficient (Ser 1926, Sondow 2003b, Guillera and Sondow 2005). Another proof of product (55) as well as an explanation for the resemblance between this product and the Wallis formula-like "faster product for pi"

pi/2=(2/1)^(1/2)((2^2)/(1·3))^(1/4)((2^3·4)/(1·3^3))^(1/8)((2^4·4^4)/(1·3^6·5))^(1/16)...
(57)

(Guillera and Sondow 2005, Sondow 2005), is given in Sondow (2004). (This resemblance which is made even clearer by changing n->n+1 in (57).) Both these formulas are also analogous to the product for e given by

e=(2/1)^(1/1)((2^2)/(1·3))^(1/2)((2^3·4)/(1·3^3))^(1/3)((2^4·4^4)/(1·3^6·5))^(1/4)...
(58)

due to Guillera (Sondow 2005).

EulerMascheroniSondow

The values r(n) obtained after inclusion of the first n terms of the product for e^gamma are plotted above.

A curious sum limit converging to gamma is given by

lim_(n->infty)1/nsum_(k=1)^(n-1)([n/k]-n/k)=gamma
(59)

(Havil 2003, p. 113), where [x] is the ceiling function.

See also

Euler-Mascheroni Constant Approximations, Euler-Mascheroni Constant Continued Fraction, Euler-Mascheroni Constant Digits, Euler Product, Hadjicostas's Formula, Jeep Problem, Mertens Theorem, Stieltjes Constants Explore this topic in the MathWorld classroom

Related Wolfram sites

http://functions.wolfram.com/Constants/EulerGamma/

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References

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Euler-Mascheroni Constant

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Weisstein, Eric W. "Euler-Mascheroni Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MascheroniConstant.html

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