Expanding the Riemann zeta function about 
gives
 |
(1)
|
(Havil 2003, p. 118), where the constants
![gamma_n=lim_(m->infty)[sum_(k=1)^m((lnk)^n)/k-((lnm)^(n+1))/(n+1)]](/images/equations/StieltjesConstants/NumberedEquation2.svg) |
(2)
|
are known as Stieltjes constants.
Another sum that can be used to define the constants is
 |
(3)
|
These constants are returned by the Wolfram Language function StieltjesGamma[n].
A generalization 
takes 
as the coefficient of 
is the Laurent series
of the Hurwitz zeta function 
about 
. These generalized Stieltjes constants are implemented in
the Wolfram Language as StieltjesGamma[n,
a].
The case 
gives the usual Euler-Mascheroni constant
 |
(4)
|
A limit formula for 
is given by
![gamma_1=-lim_(y->infty)y{y+I[zeta(1+i/y)]},](/images/equations/StieltjesConstants/NumberedEquation5.svg) |
(5)
|
where ![I[z]](/images/equations/StieltjesConstants/Inline9.svg)
is the imaginary part and 
is the Riemann zeta
function.
An alternative definition is given by absorbing the coefficient of 
into the constant,
 |
(6)
|
(e.g., Hardy 1912, Kluyver 1927).
The Stieltjes constants are also given by
![gamma_n=lim_(z->1)[(-1)^nzeta^((n))(z)-(n!)/((z-1)^(n+1))].](/images/equations/StieltjesConstants/NumberedEquation7.svg) |
(7)
|
Plots of the values of the Stieltjes constants as a function of 
are illustrated above (Kreminski). The first few numerical
values are given in the following table.
 | OEIS |  |
| 0 | A001620 | 0.5772156649 |
| 1 | A082633 |  |
| 2 | A086279 |  |
| 3 | A086280 | 0.002053834420 |
| 4 | A086281 | 0.002325370065 |
| 5 | A086282 | 0.0007933238173 |
Briggs (1955-1956) proved that there infinitely many 
of each sign. The signs of
for 
,
1, ... are 1, 
,
,
1, 1, 1, 
,
,
,
,
... (OEIS A114523), and the run lengths of
consecutive signs are 1, 2, 3, 4, 3, 4, 5, 4, 5, 5, 5, ... (OEIS A114524).
A plot of run lengths is shown above.
Berndt (1972) gave upper bounds of
 |
(8)
|
However, these bounds are extremely weak. A stronger bound is given by
 |
(9)
|
for 
(Matsuoka 1985).
Vacca (1910) proved that the Euler-Mascheroni constant may be expressed as
 |
(10)
|
where 
is the floor function and the lg
function 
is the logarithm to base 2. Hardy (1912) derived the
formula
 |
(11)
|
from Vacca's expression.
Kluyver (1927) gave similar series for 
valid for all 
,
 |
(12)
|
where 
is a Bernoulli polynomial. However, this
series converges extremely slowly, requiring more than 
terms to get two digits of 
and many more for higher order 
.
can also be expressed as a single sum using
 |
(13)
|
also appears in the asymptotic expansion of the sum
 |
(14)
|
where 
was called 
and given incorrectly by Ellision and Mendès-France (1975) (and the error
was subsequently reproduced by Le Lionnais 1983, p. 47). The exact form of (14) is given by
 |
(15)
|
where 
is a harmonic number and 
is a generalized Stieltjes constant.
A set of constants related to 
is
![delta_n=lim_(m->infty)[sum_(k=1)^m(lnk)^n-int_1^m(lnx)^ndx-1/2(lnm)^n]](/images/equations/StieltjesConstants/NumberedEquation16.svg) |
(16)
|
(Sitaramachandrarao 1986, Lehmer 1988).
The Stieltjes constants also satisfy the beautiful sum
![sum_(k=0)^infty(gamma_(k+n))/(k!)=(-1)^n[n!+zeta^((n))(0)]](/images/equations/StieltjesConstants/NumberedEquation17.svg) |
(17)
|
(O. Marichev, pers. comm., 2008).
." Quart. J. Pure Appl. Math. 43, 215-216,
1912.Hardy, G. H. and Wright, E. M. "The Behavior of
when 
."
§17.3 in 
is 
keine Nullstelle besitzt." Univ. Nac. Tucmán
Rev. Ser. A 16, 209-217, 1966.Le Lionnais, F.