Euler-Mascheroni Constant

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

			(1)
			(2)

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

has the numerical value

(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 to be irrational (Havil 2003, p. 52), although no written reference for this quote seems to be known. Hilbert mentioned the irrationality of 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 is a simple fraction , then it is known that (Brent 1977; Wells 1986, p. 28), which was subsequently improved by T. Papanikolaou to (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 is given by 2, 7, 13, 19, 85, 2601, 9602, 46268, 4812284, ... (OEIS A053977).

The Euler-Mascheroni constant arises in many integrals

			(4)
			(5)
			(6)
			(7)

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

			(8)
			(9)

Double integrals include

(10)

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

			(11)
			(12)
			(13)

(OEIS A094640; Sondow 2005).

is also given by Mertens theorem

(14)

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

(15)

It is also given by series

(16)

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

(17)

and then substituting the telescoping sum

(18)

for , which is its sum since again

(19)

obtaining

			(20)
			(21)

which equals equation (◇).

Other series include

			(22)
			(23)

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

(24)

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

(25)

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

(26)

and add

(27)

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

The sums

			(28)
			(29)

(Gosper 1972, with replacing the undefined ; 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, 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

(30)

To see the equivalence, expand in a geometric series, multiply by , and integrate termwise (Sondow and Zudilin 2003).

Other series for include

(31)

(Flajolet and Vardi 1996), and

(32)

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

A rapidly converging limit for is given by

			(33)
			(34)

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

Another limit formula is given by

(35)

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

(36)

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

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

(37)

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

An elegant identity for is given by

(38)

where is a modified Bessel function of the first kind, is a modified Bessel function of the second kind, and

(39)

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

			(40)
			(41)
			(42)
			(43)

with , , , and (Borwein and Bailey 2003, pp. 138-139).

Reformulating this identity gives the limit

(44)

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

Infinite products involving also arise from the Barnes G-function with positive integer . The cases and give

			(45)
			(46)

The Euler-Mascheroni constant is also given by the expressions

			(47)
			(48)

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

(49)

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

(50)

(Sondow 1998), and

(51)

(Le Lionnais 1983).

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

(52)

where is the floor function, and satisfies the inequality

(53)

(Young 1991).

The symbol is sometimes also used for

(54)

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

There is a the curious radical representation

(55)

which is related to the double series

(56)

and 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 "

(57)

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