Collection of formulae for Euler’s constant γ Xavier Gourdon and Pascal Sebah numbers.computation.free.fr/Constants/constants.html November 11, 2008
1
Integral formulae
Euler’s constant γ appears in many integrals (often related, for example, to the gamma function or the logarithmic integral function), we propose here to enumerate a selection of such integrals. Some of those can be deduced from others by elementary changes of variable. We use the notation bxc for the floor function and {x} for the fractional part of a real number x. Z ∞ Z ∞ t − btc {t} 1−γ = dt = dt 2 t t2 1 1 Z ∞ −γ = e−t log tdt = Γ0 (1) 0 Z ∞ π2 γ2 + = e−t log2 t dt = Γ(2) (1) (Euler-Mascheroni) 6 0 Z ∞ γπ 2 −γ 3 − − 2ζ(3) = e−t log3 t dt = Γ(3) (1) (Euler-Mascheroni) 2 0 Z 1 1 γ = − log log dt t 0 Z ∞ 2 γ + 2 log 2 = −4 π −1/2 e−t log t dt ¶ µ 0 Z ∞ 1 1 −t γ = e − dt 1 − e−t t 0 ¶2 Z ∞µ 1 1 1 log 2π − γ − = − dt ([4]) 2 et − 1 t 0 ¶ Z 1µ 1 1 + dt γ = t log(1 − t) 0 ¶ Z ∞µ 1 dt γ = − e−t 1 + t t 0 ¶ Z ∞µ 1 dt γ = − cos t 2 1 + t t 0 1
Z γ
∞
= −∞
Z
(α − β)γ
=
αβ Z
γ γ γ γ γ γ γ γ
log(1 + e−t )et dt t2 + π 2 ∞
α
(Pr´evost [11])
β
e−t − e−t dt t
α > 0, β > 0
0 1Z 1
1−x dxdy (Sondow [15]) (1 − xy) log(xy) 0 0 Ã∞ ! Z 1 X k 1 2 = 1− t dt (Catalan) 0 1+t k=1 Ã∞ ! Z 1 X k 1 + 2t = 1− t3 dt (Ramanujan [13]) 2 0 1+t+t k=1 Z ∞ 1 t dt = +2 (Hermite) 2 + 1)(e2πt − 1) 2 (t 0 Z ∞ 1 1 2t dt 1 + − log n + = 1 + + ··· + 2 2 2 n − 1 2n (t + n )(e2πt − 1) 0 Z 1 1 − e−t − e−1/t = (Barnes [1]) dt t 0 Z x Z ∞ 1 − cos t cos t = dt − dt − log x x>0 t t 0 x Z x Z ∞ −t 1 − e−t e = dt − dt − log x x>0 t t 0 x =
−
This last integral is often used to deduce an efficient algorithm to compute many digits of γ (see [6]).
2
Series formulae
In this section we provide a list of various series for γ.
2.1
Basic series Ã
γ
=
γ
=
γ
=
γ
=
! n X 1 lim (Euler) − log n n→∞ k k=1 µ ¶¶ Xµ1 1 1+ + log 1 − (Euler) k k k≥2 µ ¶ 2k + 2 π X 1 − 2 log log + 4 k 2k + 1 k≥1 à n ! X1 1 lim − log (n(n + 1)) (Cesaro) n→∞ k 2 k=1
2
Ã
γ
=
γ
=
γ
=
γ
=
γ
=
γ
=
γ
=
! 2 lim − log(4n) n→∞ 2k − 1 k=1 à n µ ¶! X1 1 1 lim − log n2 + n + n→∞ k 2 3 k=1 à n õ !! ¶2 X1 1 1 1 2 lim − log n +n+ − n→∞ k 4 3 45 k=1 µ µ ¶ ¶ (n − 1) (n − 1) (n − 2) + lim 2 1 + + · · · − log(2n) n→∞ 2n 3n2 µ ¶ X 1 1 lim+ − k (Sondow [14]) s k s s→1 k≥1 µ µ ¶¶ 1 (Demys) lim n − Γ n→∞ n log k log 2 1 X (−1)k + 2 log 2 k n X
(Kruskal [9])
k≥2
The last alternating series may be convenient to estimate Euler’s constant to thousand decimal places thanks to convergence acceleration of alternating series (see the related essay at [6]). 2.1.1
Ramanujan’s approach
In Ramanujan’s famous notebooks, we find another kind of Euler-Maclaurin like asymptotic expansion; he writes n X 1 1 1 1 1 1 − log (n(n + 1)) ≈ γ + − + − k 2 12p 120p2 630p3 1680p4
(1)
k=1
with the variable p = 21 n(n + 1), which extends Cesaro’s estimation. This representation may also be deduced from the classical Euler-Maclaurin expansion with Bernoulli’s numbers.
2.2
Around the zeta function
When he studied γ, Euler found some interesting series which allow to compute it with the integral values of the Riemann zeta function. He used one of those to give the first estimation of his constant (a five correct digits approximation). There are many formulae giving γ as function of the Riemann zeta function ζ(s), some are easy to prove. We provide the demonstration of one example. By definition, we may write ! à n µ ¶¶ Xµ1 X1 k−1 − log n = 1 + + log γ = lim n→∞ k k k k≥2
k=1
3
= 1+
¶¶ µ 1 + log 1 − k k
Xµ1 k≥2
and using the series for log(1 − x) when x = k1 < 1 gives X X 1 γ =1− `k ` k≥2
`≥2
then by associativity of this positive sum X1 X 1 X1 =1− (ζ(`) − 1) . γ =1− ` ` k ` `≥2
k≥2
`≥2
So we have ve just demonstrated a first relation between γ and the zeta functions. Because it is clear that ζ(`) − 1 is equivalent to 1/2` when ` becomes large, some of those series have geometric convergence (of course one has to evaluate ζ(`) for different integral values of `). A general improvement can be made if we start the series with k > 2 by computing its first terms, that is, for any integer n > 1: µ ¶¶ µ ¶¶ n µ X X µ1 1 k−1 k−1 γ =1+ + log + + log k k k k k=2
k≥n+1
and the result now becomes µ ¶¶ X µ ¶ n µ X 1 k−1 1 1 1 γ =1+ + log − ζ(`) − 1 − ` − · · · − ` , k k ` 2 n k=2
`≥2
and this time ζ(`, n + 1) = ζ(`) − 1 −
1 1 1 − ··· − ` ∼ 2` n (n + 1)`
so that the rate of convergence is better. This function ζ(s, a) is known as the Hurwitz Zeta function. For different values of n, the identity for γ gives ¶ X1µ 3 1 n = 2 γ = − log 2 − ζ(`) − 1 − ` 2 ` 2 `≥2 µ ¶ X1 1 1 11 − log 3 − ζ(`) − 1 − ` − ` n = 3 γ= 6 ` 2 3 `≥2 ... or in term of ζ(s, a) and the harmonic number Hn γ = Hn − log n −
X ζ(`, n + 1) `≥2
4
`
.
(2)
2.2.1
Zeta series γ
= 1−
X ζ(k) − 1
γ
X (k − 1)(ζ(k) − 1)
=
(Euler)
k
k≥2
γ
(Euler)
k
k≥2
= 1−
log 2 X ζ(2k + 1) − 1 − 2 2k + 1 k≥1
γ
X ζ(2k + 1) − 1
= log 2 −
k+1 µ ¶ X ζ(2k + 1) − 1 3 = 1 − log − 2 4k (2k + 1) k≥1
γ
(Euler-Stieltjes)
k≥1
γ
= 2 − 2 log 2 −
X ζ(2k + 1) − 1 (k + 1)(2k + 1)
(Glaisher)
k≥1
γ
X
=
(−1)k
k≥2
γ
ζ(k) k
= 1 − log 2 +
(Euler)
X
(−1)k
k≥2
γ
=
ζ(k) − 1 k
X 3 ζ(k) − 1 − log 2 − (−1)k (k − 1) 2 k
(Flajolet-Vardi)
k≥2
γ
=
ζ(k) − 1 5 1X (−1)k (k − 2) − log 2 − 4 2 k k≥3
γ
= log(8π) − 3 + 2 µ
γ
2.3 γ
= 1 + log
=
X
=
`
2`+1 X−1 k=2`
log 2 −
X `≥1
γ
(−1)k
k≥2
+2
X
ζ(k) − 1 k+1
(−1)k
k≥2
ζ(k) − 1 2k k
Other series
`≥1
γ
16 9π
¶
X
=
X k≥1
(−1)k
(−1)k k 1 2
2`
(Vacca [17], Franklin [5])
3` −1) (X
1 3
k= 21 (3`−1 +1)
blog2 kc k
(3k) − 3k
(Vacca [17])
5
(Ramanujan [2])
à 1−γ
=
γ
=
lim
n→∞
X ak k≥1
γ
=
! n 1 X nno n k (Kluyver)
k
1 − log 2 +
X k≥1
γ
=
n−1 X k=1
(de la Vall´ee Poussin [18])
k=1
ak k(k + 1)
(Kluyver)
X 1 − log n + (n − 1)! k
µ
k≥1
ak k(k + 1) · · · (k + n − 1)
¶ (Kluyver [8])
In Kluyver’s formulae the ak are rational numbers defined by:
a1 = and 0 < ak ≤
1 k+1 .
1 , 2
k−1
ak =
1 X k−` ak−` k+1 `(` + 1) `=1
Here are the first values:
1 1 1 19 3 863 275 ,a = ,a = ,a = ,a = ,a = ,a = . 2 2 12 3 24 4 720 5 160 6 60480 7 24192 Kluyver’s last relation may be used to compute a few thousand digits of γ. a1 =
3 3.1
Euler’s constant and number theory Dirichlet estimation
In 1838, Lejeune Dirichlet (1805-1859) showed that the mean of the divisors function d(k) (numbers of divisors of k, [7]) of all integers from 1 to n is such as µ ¶ n 1X 1 d(k) = log n + 2γ − 1 + O √ . n n k=1
For example, a direct computation with n = 105 produces n
1X d(k) − log n = 0.1545745350... n k=1
while 2γ − 1 = 0.1544313298....
3.2
Mertens formulae
If p represents a prime number, Franz Mertens (1840-1927) gave in 1874 the two beautiful formulae ([10], [7]): µ ¶−1 1 1 Y 1− (3) eγ = lim n→∞ log n p p≤n
6
6eγ π2
=
¶ µ 1 Y 1 1+ n→∞ log n p lim
(4)
p≤n
The product (3) is equivalent to the series µ ¶ X 1 γ = lim − log 1 − − log log n n→∞ p
(5)
p≤n
but when p is large µ ¶ µ ¶ 1 1 1 − log 1 − = +O p p p2 and the relation (5) for γ is very similar to its definition relation, but this time, only the prime numbers are taken into account in the sum.
3.3
Von Mangoldt function
The von Mangoldt function Λ(k) is generated by mean of the Zeta function as follow [7]: −
ζ 0 (s) X Λ(k) = , ζ(s) ks
s>1
(6)
k≥1
and it is also defined by ½ Λ(k) = log p if k = pm for any prime p, Λ(k) = 0 otherwise. The relation (6) may also be written as ζ(s) +
X Λ(k) − 1 ζ 0 (s) =− , ζ(s) ks
s>1
k≥1
from which, by taking the limits as s tends to 1, we deduce the interesting series expansion: γ=−
1 X Λ(k) − 1 . 2 k
(7)
k≥1
It is a very slow and irregular converging series, partial sums Sn with n terms are S1,000 S10,000
= 0.57(835...), = 0.57(648...),
S100,000
= 0.57(694...), = 0.577(417...).
S1,000,000
7
4
Approximations
Unlike the constant π, few approximations are available for γ, it may be useful to list a few of those.
4.1
Rational approximations
The continued fraction representation makes it easy to find the sequence of the best rational approximations: γ = [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, ...], that is, in term of fractions ·
¸ 1 3 4 11 15 71 228 3035 15403 18438 33841 289166 323007 0, 1, , , , , , , , , , , , , , ... . 2 5 7 19 26 123 395 5258 26685 31943 58628 500967 559595
For example, by mean of the continued fractions, we get the two approximative values ¯ ¯ ¯ 33841 ¯ −11 ¯ ¯ ¯ 58628 − γ ¯ < 3.2 × 10 and ¯ ¯ ¯ 376566901 ¯ −19 ¯ ¯ ¯ 652385103 − γ ¯ < 2.0 × 10 . A more exotic fraction due to Castellanos [3] is ¯ ¯ ¯ ¯ 9903 − 553 − 792 − 42 ¯ − γ ¯¯ < 3.8 × 10−15 . ¯ 5 70
4.2
Other approximations γ
≈
γ
≈
γ
≈
γ
≈
1 √ = 0.577(350...) 3 √ 41 − 1241 = 0.57721(700...) q10 √ 3 66 + 6 = 0.577215(396...) 43 √ ´ 59 ³ 1 + 11 7 = 0.577215664(894...) 3077 8
µ γ
≈ µ
γ
≈
γ
≈
γ
≈
γ
≈
γ
≈
γ
≈
7 83
¶2/9 = 0.577215(209...)
(Castellanos [3])
¶1/6 803 + 92 = 0.577215664(572...) (Castellanos [3]) 614 4 √ = 0.57721(411...) 2 3 + 5 log 2 3 = 0.5772(311...) 3 + 2 log 3 µ ¶ 73 71 log = 0.57721566(601...) 293 7 µ ¶ 16 5 + log = 0.57721566(525...) 241 3 3696 log (840) = 0.5772156649015(627...) 43115
References [1] E.W. Barnes, On the expression of Euler’s constant as a definite integral, Messenger, (1903), vol. 33, pp. 59-61 [2] B.C. Berndt and T. Huber, A fragment on Euler’s constant in Ramanujan’s lost notebook, (2007) [3] D. Castellanos, The Ubiquitous Pi. Part I., Math. Mag., (1988), vol. 61, pp. 67-98 [4] S. Finch and P. Sebah, Comment on ”Volumes spanned by random points in the hypercube”, to appear in Random Structures and Algorithms, (2008) [5] F. Franklin, On an expression for Euler’s constant, J. Hopkins circ., (1883), vol. 2, p. 143 [6] X. Gourdon and P. Sebah, Numbers, Constants and Computation, World Wide Web site at the adress: http://numbers.computation.free.fr/Constants/constants.html, (1999) [7] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, (1979) [8] J.C. Kluyver, De constante van Euler en de natuurlijhe getallen, Amst. Ak., (1924), vol. 33, pp. 149-151 [9] M.D. Kruskal, American Mathematical Monthly, (1954), vol. 61, pp. 392397 [10] F. Mertens, Journal f¨ ur Math., (1874), vol. 78, pp. 46-62 9
[11] M. Pr´evost, A Family of Criteria for Irrationality of Euler’s Constant, preprint, (2005) [12] S. Ramanujan, A series for Euler’s constant γ, Messenger, (1916), vol. 46, p. 73-80 [13] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, (1988) [14] J. Sondow, An antisymmetric formula for Euler’s constant, Mathematics Magazine, (1998), vol. 71, number 3, pp. 219-220 [15] J. Sondow, Criteria for irrationality of Euler’s constant, Proc. Amer. Math. Soc., (2003), vol. 131, pp. 3335-3344 P∞ [16] T.J. Stieltjes, Tables des valeurs des sommes Sk = n=1 n−k ,Acta Mathematica, (1887), vol. 10, pp. 299-302 [17] G. Vacca, A New Series for the Eulerian Constant, Quart. J. Pure Appl. Math, (1910), vol. 41, pp. 363-368 [18] C. de la Vall´ee Poussin, Sur les valeurs moyennes de certaines fonctions arithm´etiques, Annales de la soci´et´e scientifique de Bruxelles, (1898), vol. 22, pp. 84-90
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