Collection of formulae for Euler's constant g

1  Integral formulae

Euler's constant g 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 ë x û for the floor function and { x} for the fractional part of a number x.

1-g = ó õ ¥ 1   t- ë t û t2 dt= ó õ ¥ 1   { t} t2 dt -g = ó õ ¥ 0  e-tlog(t) dt=G¢(1) g2+  p2 6 = ó õ ¥ 0  e-tlog2(t) dt=G(2)(1)       (Euler-Mascheroni) -g3-  gp2 2 -2z(3) = ó õ ¥ 0  e-tlog3(t) dt=G(3)(1)       (Euler-Mascheroni) g = - ó õ 1 0  log( log(1/t)) dt g+2log(2) = -4 p-1/2  ó õ ¥ 0  e-t2 log(t) dt g = ó õ ¥ 0  e-t æ è  1 1-e-t -  1 t ö ø dt g = ó õ 1 0  æ è  1 t +  1 log(1-t) ö ø dt g = ó õ ¥ 0  æ è  1 1+t -e-t ö ø  dt t g = ó õ ¥ 0  æ è  1 1+t2 -cos(t) ö ø  dt t (a-b)g = ab ó õ ¥ 0   e-ta-e-tb t dt       a > 0,b > 0 g = - ó õ 1 0  ó õ 1 0   (1-x) (1-xy)log(xy) dxdy      (Sondow [12]) g = 1- ó õ 1 0   1 1+t æ è ¥ å k=1  t2k ö ø  dt       (Catalan) g = 1- ó õ 1 0   1+2t 1+t+t2 æ è ¥ å k=1  t3k ö ø  dt       (Ramanujan [10]) g =  1 2 +2 ó õ ¥ 0   t dt (t2+1)(e2pt-1)        (Hermite) g = 1+  1 2 +...+  1 n-1 +  1 2n -log(n)+2 ó õ ¥ 0   t dt (t2+n2)(e2pt-1) g = ó õ 1 0   1-e-t-e-1/t t  dt       (Barnes[1]) g = ó õ x 0   1-cos(t) t dt- ó õ ¥ x   cos(t) t dt-log(x)       x > 0 g = ó õ x 0   1-e-t t dt- ó õ ¥ x   e-t t dt-log(x)       x > 0

This last integral is often used to deduce an efficient algorithm to compute many digits of g (see [4]).

2  Series formulae

Now here is a list of series for g.

2.1  Basic series

g = lim n® ¥  æ è n å k=1   1 k -log(n) ö ø        (Euler) g = 1+ ¥ å k=2  æ è  1 k +log æ è 1-  1 k ö ø ö ø        (Euler) g = lim n® ¥  æ è n å k=1   1 k -  1 2 log( n(n+1)) ö ø        (Cesaro) g = lim n® ¥  æ è n å k=1   2 2k-1 -log(4n) ö ø g = lim n® ¥  æ è n å k=1   1 k -  1 2 log æ è n2+n+  1 3 ö ø ö ø g = lim n® ¥  æ è n å k=1   1 k -  1 4 log æ è æ è n2+n+  1 3 ö ø 2   -  1 45 ö ø ö ø g = lim n® ¥  æ è 2 æ è 1+  (n-1) 2n +  ( n-1) ( n-2) 3n2 +... ö ø -log(2n) ö ø        (Kruskal [7]) g = lim s® 1+  æ è ¥ å k=1  æ è  1 ks -  1 sk ö ø ö ø        (Sondow [11]) g = lim n® ¥  æ è n-G æ è  1 n ö ø ö ø        (Demys) g =  log(2) 2 +  1 log(2) ¥ å k=2  (-1)k  log(k) k

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 [4]).

2.1.1  Ramanujan's approach

In Ramanujan's famous notebooks, we find another kind of Euler-Maclaurin like asymptotic expansion; he writes

n å k=1   1 k -  1 2 log( n(n+1)) » g+  1 12p -  1 120p2 +  1 630p3 -  1 1680p4

with the variable p=n(n+1)/2, 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 g, 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 g as function of the Riemann zeta function z(s), some are easy to prove. We provide the demonstration of one example.

By definition, we may write:

g = lim n® ¥  æ è n å k=1   1 k -log(n) ö ø =1+ ¥ å k=2  æ è  1 k +log æ è  k-1 k ö ø ö ø =1+ ¥ å k=2  æ è  1 k +log æ è 1-  1 k ö ø ö ø

and using the series for log(1-x) when x=1/k < 1:

g = 1- ¥ å k=2  æ è ¥ å i=2   1 iki ö ø

then by associativity of this positive sum:

g = 1- ¥ å i=2   1 i æ è ¥ å k=2   1 ki ö ø =1- ¥ å i=2   1 i ( z(i)-1) .

So we've just demonstrated a first relation between g and the zeta functions. Because it's clear that z(i)-1 is equivalent to 1/2ⁱ when i becomes large, some of those series have geometric convergence (of course one has to evaluate z(i) for different integral values of i).

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:

g = 1+ n å k=2  æ è  1 k +log æ è  k-1 k ö ø ö ø + ¥ å k=n+1  æ è  1 k +log æ è  k-1 k ö ø ö ø

and the result now becomes

g = 1+ n å k=2  æ è  1 k +log æ è  k-1 k ö ø ö ø - ¥ å i=2   1 i æ è z(i)-1-  1 2i -...-  1 ni ö ø ,

and this time

z(i,n+1)=z(i)-1-  1 2i -...-  1 ni ~  1 (n+1)i

so that the rate of convergence is better. This function z(s,a) is known as the Hurwitz Zeta function. For different values of n, the identity for g gives

n = 2       g =  3 2 -log(2)- ¥ å i=2   1 i æ è z(i)-1-  1 2i ö ø n = 3       g =  11 6 -log(3)- ¥ å i=2   1 i æ è z(i)-1-  1 2i -  1 3i ö ø ...

or in term of z(s,a) and the harmonic number H_n

g = Hn-log(n)- ¥ å i=2   z(i,n+1) i .

2.2.1  Zeta series

g = 1- ¥ å k=2   (z(k)-1) k        (Euler) g = ¥ å k=2   (k-1)(z(k)-1) k        (Euler) g = 1-  log(2) 2 - ¥ å k=1   (z(2k+1)-1) 2k+1 g = log(2)- ¥ å k=1   z(2k+1) 4k(2k+1)        (Euler) g = 1-log æ è  3 2 ö ø - ¥ å k=1   (z(2k+1)-1) 4k(2k+1)        (Euler-Stieltjes) g = 1- ¥ å k=1   z(2k+1) (k+1)(2k+1)        (Glaisher) g = 2-2log(2)- ¥ å k=1   (z(2k+1)-1) (k+1)(2k+1)        (Glaisher) g = ¥ å k=2  (-1)k  z(k) k        (Euler) g = 1-log(2)+ ¥ å k=2  (-1)k  (z(k)-1) k g =  3 2 -log(2)- ¥ å k=2  (-1)k(k-1)  (z(k)-1) k        (Flajolet-Vardi) g =  5 4 -log(2)-  1 2 ¥ å k=3  (-1)k(k-2)  (z(k)-1) k g = log(8p)-3+2 ¥ å k=2  (-1)k  (z(k)-1) k+1 g = log æ è  4 p ö ø +2 ¥ å k=2  (-1)k  z(k) 2kk g = 1+log æ è  16 9p ö ø +2 ¥ å k=2  (-1)k  (z(k)-1) 2kk

2.3  Other series

g = ¥ å i=1  i æ è 2i+1-1 å k=2i   (-1)k k ö ø        (Vacca [14], Franklin [3]) g = ¥ å k=1  (-1)k   ë log2k û k        (Vacca [14]) 1-g = lim n® ¥  æ è  1 n n å k=1  ì í î  n k ü ý þ ö ø        (de la Vallée Poussin [15]) g = ¥ å k=1   ak k        (Kluyver) g = 1-log(2)+ ¥ å k=1   ak k(k+1)        (Kluyver) g = n-1 å k=1   1 k -log(n)+(n-1)! ¥ å k=1  æ è  ak k(k+1)...(k+n-1) ö ø       (Kluyver [6])

In Kluyver's formulae the a_k are rational numbers defined by:

a1 =  1 2 ak =  1 k+1 k-1 å i=1   k-i i(i+1) ak-i

and 0 < a_k £ 1/(k+1). Here are the first values:

a1=  1 2 ,a2=  1 12 ,a3=  1 24 ,a4=  19 720 ,a5=  3 160 ,a6=  863 60480 ,a7=  275 24192 ,...

Kluyver's last relation may be used to compute a few thousand digits of g.

3  Euler's constant and number theory

3.1  Dirichlet estimation

In 1838, Lejeune Dirichlet (1805-1859) showed that the mean of the divisors function d(k) (numbers of divisors of k, [5]) of all integers from 1 to n is such as

1 n n å k=1  d(k)=logn+(2g-1)+O æ è  1 Ön ö ø .

For example a direct computation with n=10⁵ produces

1 n n å k=1  d(k)-logn=0.1545745350...

while ( 2g-1) = 0.1544313298...

3.2  Mertens formulae

If p represents a prime number, Franz Mertens (1840-1927) gave in 1874 the two beautiful formulae ([8], [5]):

eg = lim n® ¥   1 log(n) Õ p £ n  æ è 1-  1 p ö ø -1   (1)  6eg p2 = lim n® ¥   1 log(n) Õ p £ n  æ è 1+  1 p ö ø (2)

The product (1) is equivalent to the series

g = lim n® ¥  æ è å p £ n  -log æ è 1-  1 p ö ø -log( log(n)) ö ø

(3)

but when p is large

-log æ è 1-  1 p ö ø =  1 p +O æ è  1 p2 ö ø

and the relation (3) for g is very similar to it's 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 L (k) is generated by mean of the Zeta function as follow [5]:

-  z¢(s) z(s) = ¥ å k=1   L(k) ks        s > 1

(4)

and it's also defined by

ì í î L (k)=log(p) if k=pm for any prime  p L (k)=0 otherwise.

The relation (4) may also be written as

z(s)+  z¢(s) z(s) =- ¥ å k=1   ( L (k)-1) ks        s > 1

from which, by taking the limits as s tends to 1, we deduce the interesting series expansion:

g = -  1 2 ¥ å k=2   (L (k)-1) k .

4  Approximations

Unlike the constant p, few approximations are available for g, 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:

g = [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

é ë 0,1,  1 2 ,  3 5 ,  4 7 ,  11 19 ,  15 26 ,  71 123 ,  228 395 ,  3035 5258 ,  15403 26685 ,  18438 31943 ,  33841 58628 ,  289166 500967 ,  323007 559595 ,... ù û .

For example, by mean of the continued fractions, we get

ê ê  33841 58628 -g ê ê < 3.2×10-11

and

ê ê  376566901 652385103 -g ê ê < 2.0×10-19.

A more exotic fraction due to Castellanos [2] is

ê ê  9903-553-792-42 705 -g ê ê < 3.8.10-15.

4.2  Other approximations

g »  1 Ö3 =0.577(350...) g » 41- Ö 1241 10 =0.57721(700...) g »  3 43 Ö   66+Ö6   =0.577215(396...) g »  59 3077 ( 1+11Ö7) = 0.577215664(894...) g » æ è  7 83 ö ø 2/9   =0.577215(209...)      (Castellanos [2]) g » æ è  803+92 614 ö ø 1/6   =0.577215664(572...)       (Castellanos [2]) g »  4 2Ö3+5log2 =0.57721(411...) g »  3 3+2log3 =0.5772(311...) g »  73 293 log æ è  71 7 ö ø =0.57721566(601...) g »  3696 43115 log( 840) = 0.5772156649015(627...)

References

[1]

E.W. Barnes, On the expression of Euler's constant as a definite integral, Messenger, (1903), vol. 33, p. 59-61

[2]

D. Castellanos, The Ubiquitous Pi. Part I., Math. Mag., (1988), vol. 61, p. 67-98

[3]

F. Franklin, On an expression for Euler's constant, J. Hopkins circ., (1883), vol. 2, p. 143

[4]

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)

[5]

G.H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, (1979)

[6]

J. C. Kluyver, De constante van Euler en de natuurlijhe getallen, Amst. Ak., (1924), vol. 33, p. 149-151

[7]

M.D. Kruskal, American Mathematical Monthly, (1954), vol. 61, p. 392-397

[8]

F. Mertens, Journal für Math., (1874), vol. 78, p. 46-62

[9]

S. Ramanujan, A series for Euler's constant g, Messenger, (1916), vol. 46, p. 73-80

[10]

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, (1988)

[11]

J. Sondow, An Antisymmetric Formula for Euler's Constant, Mathematics Magazine, (1998), vol. 71, number 3, p. 219-220

[12]

J. Sondow, Criteria for Irrationality of Euler's Constant, Submitted to Proc. Amer. Math. Soc., (2002)

[13]

T. J. Stieltjes, Tables des valeurs des sommes S_k=å_n=1^¥n^-k,Acta Mathematica, (1887), vol. 10, p. 299-302

[14]

G. Vacca, A New Series for the Eulerian Constant, Quart. J. Pure Appl. Math, (1910), vol. 41, p. 363-368

[15]

C. de la Vallée Poussin, Sur les valeurs moyennes de certaines fonctions arithmétiques, Annales de la société scientifique de Bruxelles, (1898), vol. 22, p. 84-90