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
=



1 
 t- ë t û

t2
dt=


1 
 { t}

t2
dt
-g
=



0 
e-tlog(t) dt=G(1)
g2+  p2

6
=



0 
e-tlog2(t) dt=G(2)(1)       (Euler-Mascheroni)
-g3-  gp2

2
-2z(3)
=



0 
e-tlog3(t) dt=G(3)(1)       (Euler-Mascheroni)
g
=
-
1

0 
log( log(1/t)) dt
g+2log(2)
=
-p-1/2 


0 
e-t2 log(t) dt
g
=



0 
e-t
 1

1-e-t
-  1

t

dt
g
=

1

0 

 1

t
+  1

log(1-t)

dt
g
=



0 

 1

1+t
-e-t
 dt

t
g
=



0 

 1

1+t2
-cos(t)
 dt

t
(a-b)g
=
ab


0 
 e-ta-e-tb

t
dt       a > 0,b > 0
g
=
-
1

0 

1

0 
 (1-x)

(1-xy)log(xy)
dxdy      (Sondow [12])
g
=
1-
1

0 
 1

1+t



k=1 
t2k
 dt       (Catalan)
g
=
1-
1

0 
 1+2t

1+t+t2



k=1 
t3k
 dt       (Ramanujan [10])
g
=
 1

2
+2


0 
 t dt

(t2+1)(e2pt-1)
       (Hermite)
g
=
1+  1

2
+...+  1

n-1
+  1

2n
-log(n)+2


0 
 t dt

(t2+n2)(e2pt-1)
g
=

1

0 
 1-e-t-e-1/t

t
 dt       (Barnes[1])
g
=

x

0 
 1-cos(t)

t
dt-


x 
 cos(t)

t
dt-log(x)       x > 0
g
=

x

0 
 1-e-t

t
dt-


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  

n

k=1 
 1

k
-log(n)
       (Euler)
g
=
1+

k=2 

 1

k
+log
1-  1

k


       (Euler)
g
=

lim
n  

n

k=1 
 1

k
-  1

2
log( n(n+1))
       (Cesaro)
g
=

lim
n  

n

k=1 
 2

2k-1
-log(4n)
g
=

lim
n  

n

k=1 
 1

k
-  1

2
log
n2+n+  1

3


g
=

lim
n  

n

k=1 
 1

k
-  1

4
log

n2+n+  1

3

2

 
-  1

45


g
=

lim
n  

2
1+  (n-1)

2n
+  ( n-1) ( n-2)

3n2
+...
-log(2n)
       (Kruskal [7])
g
=

lim
s 1+ 



k=1 

 1

ks
-  1

sk


       (Sondow [11])
g
=

lim
n  

n-G
 1

n


       (Demys)
g
=
 log(2)

2
+  1

log(2)


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

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  

n

k=1 
 1

k
-log(n)
=1+

k=2 

 1

k
+log
 k-1

k


=1+

k=2 

 1

k
+log
1-  1

k


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


g = 1-

k=2 



i=2 
 1

iki

then by associativity of this positive sum:


g = 1-

i=2 
 1

i



k=2 
 1

ki

=1-

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

k=2 

 1

k
+log
 k-1

k


+

k=n+1 

 1

k
+log
 k-1

k


and the result now becomes
g = 1+ n

k=2 

 1

k
+log
 k-1

k


-

i=2 
 1

i

z(i)-1-  1

2i
-...-  1

ni

,
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)-

i=2 
 1

i

z(i)-1-  1

2i

n
=
3       g =  11

6
-log(3)-

i=2 
 1

i

z(i)-1-  1

2i
-  1

3i

...      
or in term of z(s,a) and the harmonic number Hn
g = Hn-log(n)-

i=2 
 z(i,n+1)

i
.

2.2.1  Zeta series


g
=
1-

k=2 
 (z(k)-1)

k
       (Euler)
g
=


k=2 
 (k-1)(z(k)-1)

k
       (Euler)
g
=
1-  log(2)

2
-

k=1 
 (z(2k+1)-1)

2k+1
g
=
log(2)-

k=1 
 z(2k+1)

4k(2k+1)
       (Euler)
g
=
1-log
 3

2

-

k=1 
 (z(2k+1)-1)

4k(2k+1)
       (Euler-Stieltjes)
g
=
1-

k=1 
 z(2k+1)

(k+1)(2k+1)
       (Glaisher)
g
=
2-2log(2)-

k=1 
 (z(2k+1)-1)

(k+1)(2k+1)
       (Glaisher)
g
=


k=2 
(-1)k  z(k)

k
       (Euler)
g
=
1-log(2)+

k=2 
(-1)k  (z(k)-1)

k
g
=
 3

2
-log(2)-

k=2 
(-1)k(k-1)  (z(k)-1)

k
       (Flajolet-Vardi)
g
=
 5

4
-log(2)-  1

2


k=3 
(-1)k(k-2)  (z(k)-1)

k
g
=
log(8p)-3+2

k=2 
(-1)k  (z(k)-1)

k+1
g
=
log
 4

p

+2

k=2 
(-1)k  z(k)

2kk
g
=
1+log
 16

9p

+2

k=2 
(-1)k  (z(k)-1)

2kk

2.3  Other series


g
=


i=1 
i
2i+1-1

k=2i 
 (-1)k

k

       (Vacca [14], Franklin [3])
g
=


k=1 
(-1)k   ë log2k û

k
       (Vacca [14])
1-g
=

lim
n  

 1

n
n

k=1 


 n

k



       (de la Vallée Poussin [15])
g
=


k=1 
 ak

k
       (Kluyver)
g
=
1-log(2)+

k=1 
 ak

k(k+1)
       (Kluyver)
g
=
n-1

k=1 
 1

k
-log(n)+(n-1)!

k=1 

 ak

k(k+1)...(k+n-1)

      (Kluyver [6])

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


a1
=
 1

2
ak
=
 1

k+1
k-1

i=1 
 k-i

i(i+1)
ak-i
and 0 < ak 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

k=1 
d(k)=logn+(2g-1)+O
 1

n

.

For example a direct computation with n=105 produces
 1

n
n

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)


p n 

1-  1

p

-1

 
(1)
 6eg

p2
=

lim
n  
 1

log(n)


p n 

1+  1

p

(2)

The product (1) is equivalent to the series


g =
lim
n  



p n 
-log
1-  1

p

-log( log(n))
(3)
but when p is large


-log
1-  1

p

=  1

p
+O
 1

p2

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)
=

k=1 
 L(k)

ks
       s > 1
(4)
and it's also defined by


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)
=-

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


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



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
,...
.

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

 33841

58628
-g
< 3.2×10-11
and

 376566901

652385103
-g
< 2.0×10-19.

A more exotic fraction due to Castellanos [2] is

 9903-553-792-42

705
-g
< 3.8.10-15.

4.2  Other approximations


g
 1

3
=0.577(350...)
g
41-


1241

10
=0.57721(700...)
g
 3

43


 

66+6
 
=0.577215(396...)
g
 59

3077
( 1+117) = 0.577215664(894...)
g

 7

83

2/9

 
=0.577215(209...)      (Castellanos [2])
g

 803+92

614

1/6

 
=0.577215664(572...)       (Castellanos [2])
g
 4

23+5log2
=0.57721(411...)
g
 3

3+2log3
=0.5772(311...)
g
 73

293
log
 71

7

=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 Sk=n=1n-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




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On 25 Oct 2002, 16:34.