The Stieltjes constants are a sequence of real numbers that are determined by the limit
γ
n
{\ displaystyle \ gamma _ {n}}
γ
n
: =
lim
N
→
∞
(
∑
k
=
1
N
log
n
k
k
-
log
n
+
1
N
n
+
1
)
,
n
=
0
,
1
,
2
,
...
{\ displaystyle \ gamma _ {n}: = \ lim _ {N \ to \ infty} \ left (\ sum _ {k = 1} ^ {N} {\ frac {\ log ^ {n} k} {k }} - {\ frac {\ log ^ {n + 1} N} {n + 1}} \ right), \ quad n = 0,1,2, \ dotsc}
are defined, where is Euler's constant . It is believed that they are irrational. Proof of this has not yet been found. Because of their definition, they are sometimes referred to as generalized Euler's constants . They appear in the Laurent expansion of the Riemann zeta function
γ
0
{\ displaystyle \ gamma _ {0}}
γ
{\ displaystyle \ gamma}
γ
n
{\ displaystyle \ gamma _ {n}}
ζ
(
s
)
=
1
s
-
1
+
∑
n
=
0
∞
(
-
1
)
n
γ
n
n
!
(
s
-
1
)
n
{\ displaystyle \ zeta (s) = {\ frac {1} {s-1}} + \ sum _ {n = 0} ^ {\ infty} {\ frac {(-1) ^ {n} \ gamma _ {n}} {n!}} (s-1) ^ {n}}
and when evaluating certain specific integrals on:
∫
0
∞
log
2
x
e
x
+
1
d
x
=
(
log
2
)
(
1
3
log
2
2
+
ζ
(
2
)
-
γ
2
-
2
γ
1
)
=
1.121
192486
...
{\ displaystyle \ int \ limits _ {0} ^ {\ infty} {\ frac {\ log ^ {2} x} {e ^ {x} +1}} \, \ mathrm {d} x = (\ log 2) \, {\ big (} {\ frac {1} {3}} \ log ^ {2} 2+ \ zeta (2) - \ gamma ^ {2} -2 \ gamma _ {1} {\ big )} = 1 {,} 121192486 \ dots}
They are closely related to the numbers
τ
n
: =
∑
k
=
1
∞
(
-
1
)
k
+
1
log
n
k
k
,
n
=
0
,
1
,
2
,
...
{\ displaystyle \ tau _ {n}: = \ sum _ {k = 1} ^ {\ infty} (- 1) ^ {k + 1} {\ frac {\ log ^ {n} k} {k}} , \ quad n = 0,1,2, \ dotsc}
together. These can be calculated numerically well using an acceleration of convergence (continued averaging). The recursion applies
τ
0
=
log
2
{\ displaystyle \ tau _ {0} = \ log 2}
τ
n
=
log
n
+
1
2
n
+
1
-
∑
k
=
0
n
-
1
(
n
k
)
log
n
-
k
2
⋅
γ
k
,
n
=
1
,
2
,
...
{\ displaystyle \ tau _ {n} = {\ frac {\ log ^ {n + 1} 2} {n + 1}} - \ sum _ {k = 0} ^ {n-1} {\ binom {n } {k}} \ log ^ {nk} 2 \ cdot \ gamma _ {k}, \ qquad n = 1,2, \ dotsc}
and the explicit representation with the help of Bernoulli's numbers :
γ
n
=
-
1
n
+
1
∑
k
=
0
n
+
1
(
n
+
1
k
)
B.
n
+
1
-
k
log
n
-
k
2
⋅
τ
k
,
n
=
0
,
1
,
2
,
...
{\ displaystyle \ gamma _ {n} = - {\ frac {1} {n + 1}} \ sum _ {k = 0} ^ {n + 1} {\ binom {n + 1} {k}} B_ {n + 1-k} \ log ^ {nk} 2 \ cdot \ tau _ {k}, \ quad n = 0,1,2, \ dotsc}
From the recursion we get for the identity , i. H. for Euler's constant the alternating series
n
=
1
{\ displaystyle n = 1}
τ
1
=
1
2
log
2
2
-
γ
log
2
{\ displaystyle \ tau _ {1} = {\ tfrac {1} {2}} \ log ^ {2} 2- \ gamma \ log 2}
γ
=
1
2
log
2
+
1
log
2
∑
k
=
2
∞
(
-
1
)
k
log
k
k
=
1
2
log
2
+
∑
k
=
2
∞
(
-
1
)
k
log
2
k
k
,
{\ displaystyle \ gamma = {\ frac {1} {2}} \ log 2 + {\ frac {1} {\ log 2}} \ sum _ {k = 2} ^ {\ infty} (- 1) ^ {k} {\ frac {\ log k} {k}} = {\ frac {1} {2}} \ log 2+ \ sum _ {k = 2} ^ {\ infty} (- 1) ^ {k } {\ frac {\ log _ {2} k} {k}},}
which is very similar to Vacca's range .
The result shows an oscillating behavior with the "frequency" falling asymptotically slowly towards 0. It is known that
γ
n
{\ displaystyle \ gamma _ {n}}
lim sup
n
→
∞
ln
|
γ
n
|
n
=
ln
ln
n
{\ displaystyle \ limsup _ {n \ to \ infty} {\ frac {\ ln | \ gamma _ {n} |} {n}} = \ ln \ ln n}
applies.
Numerical values
n
Decimal expansion of γ n
OEIS
0
0.577215664901532860606512090082 ...
A001620
1
−0.0728158454836767248605863758749 ...
A082633
2
−0.00969036319287231848453038603521 ...
A086279
3
0.00205383442030334586616004654275 ...
A086280
4th
0.00232537006546730005746817017752 ...
A086281
5
0.000793323817301062701753334877444 ...
A086282
6th
−0.000238769345430199609872421841908 ...
A183141
7th
−0.000527289567057751046074097505478 ...
A183167
8th
−0.000352123353803039509602052165001 ...
A183206
9
−0.000034394774418088048177914623798 ...
A184853
10
0.000205332814909064794683722289237 ...
A184854
generalization
The following is important for the Hurwitz zeta function :
γ
n
(
a
)
: =
lim
N
→
∞
(
∑
k
=
1
N
log
n
(
k
+
a
)
k
+
a
-
log
n
+
1
(
N
+
a
)
n
+
1
)
,
n
=
0
,
1
,
2
,
...
{\ displaystyle \ gamma _ {n} (a): = \ lim _ {N \ to \ infty} \ left (\ sum _ {k = 1} ^ {N} {\ frac {\ log ^ {n} ( k + a)} {k + a}} - {\ frac {\ log ^ {n + 1} (N + a)} {n + 1}} \ right), \ quad n = 0,1,2, \ dotsc}
literature
Rick Kreminski: Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants. In: Mathematics of Computation. V. 72, no. 243, 2003, pp. 1379-1397.
Charles Knessl, Mark W, Coffey: An effective asymptotic formula for the Stieltjes Constants. In: Mathematics of Computation. V. 80, no. 273, 2010, pp. 379-386.
Web links
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