Površina modrega območja konvergira k Euler-Mascheronijevi konstanti , ki je ničta Stieltjesova konstanta.
Stieltjesove konstante (ali posplošene Eulerjeve konstante [1] ) so v matematiki števila
γ
n
{\displaystyle \gamma _{n}\,}
, ki se pojavljajo v Laurentovi vrsti za Riemannovo funkcijo ζ :
ζ
(
s
)
=
1
s
−
1
+
∑
n
=
0
∞
(
−
1
)
n
n
!
γ
n
(
s
−
1
)
n
.
{\displaystyle \zeta (s)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}\;(s-1)^{n}\!\,.}
Ničta Stieltjesova konstanta
γ
0
≡
γ
=
0
,
577215664901
…
{\displaystyle \gamma _{0}\equiv \gamma ={0},577215664901\ldots \,}
je znana kot Euler-Mascheronijeva konstanta . Konstante se imenujejo po nizozemskem matematiku Thomasu Joannesu Stieltjesu in redkeje po švicarskem matematiku, fiziku in astronomu Leonhardu Eulerju .
Stieltjes je pokazal, da so konstante dane z limito :[1] [2]
γ
n
=
lim
m
→
∞
{
∑
k
=
1
m
ln
n
k
k
−
ln
n
+
1
m
n
+
1
}
.
{\displaystyle \gamma _{n}=\lim _{m\rightarrow \infty }{\left\{\sum _{k=1}^{m}{\frac {\ln ^{n}k}{k}}-{\frac {\ln ^{n+1}\!m}{n+1}}\right\}}\!\,.}
[a]
Cauchyjeva formula za odvod vodi do integralskega izraza:
γ
n
=
(
−
1
)
n
n
!
2
π
∫
0
2
π
e
−
n
i
x
ζ
(
e
i
x
+
1
)
d
x
.
{\displaystyle \gamma _{n}={\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)\mathrm {d} x\!\,.}
Več integralskih izrazov in neskončnih vrst so v svojem delu podali Jensen , Franel, Hermite , Hardy , Ramanudžan , Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine in drugi avtorji.[3] [4] [5] [6] [7] [8] Še posebej Jensen-Franelova integralska formula, večkrat napačno pripisana Ainsworthu in Howellu, pravi, da velja:
γ
n
=
1
2
δ
n
,
0
+
1
i
∫
0
∞
d
x
e
2
π
x
−
1
{
ln
n
(
1
−
i
x
)
1
−
i
x
−
ln
n
(
1
+
i
x
)
1
+
i
x
}
,
(
n
=
0
,
1
,
2
,
…
)
,
{\displaystyle \gamma _{n}\,=\,{\frac {1}{2}}\delta _{n,0}+\,{\frac {1}{i}}\!\int _{0}^{\infty }\!{\frac {\mathrm {d} x}{e^{2\pi x}-1}}\left\{{\frac {\ln ^{n}(1-ix)}{1-ix}}-{\frac {\ln ^{n}(1+ix)}{1+ix}}\right\}\,,\qquad (n=0,1,2,\ldots )\!\,,}
kjer je
δ
n
,
k
{\displaystyle \delta _{n,k}\,}
Kroneckerjeva delta .[7] [8] Med drugimi formulami so (glej: [3] [7] [9] ):
γ
n
=
−
π
2
(
n
+
1
)
∫
−
∞
+
∞
ln
n
+
1
(
1
2
±
i
x
)
cosh
2
π
x
d
x
,
(
n
=
0
,
1
,
2
,
…
)
,
{\displaystyle \gamma _{n}\,=\,-{\frac {\pi }{2(n+1)}}\!\int _{-\infty }^{+\infty }{\frac {\ln ^{n+1}\!{\big (}{\frac {1}{2}}\pm ix{\big )}}{\cosh ^{2}\!\pi x}}\,\mathrm {d} x\,,\qquad \qquad \qquad \qquad \qquad (n=0,1,2,\ldots )\!\,,}
γ
1
=
−
[
γ
−
ln
2
2
]
ln
2
+
i
∫
0
∞
d
x
e
π
x
+
1
{
ln
(
1
−
i
x
)
1
−
i
x
−
ln
(
1
+
i
x
)
1
+
i
x
}
γ
1
=
−
γ
2
−
∫
0
∞
[
1
1
−
e
−
x
−
1
x
]
e
−
x
ln
x
d
x
.
{\displaystyle {\begin{array}{l}\displaystyle \gamma _{1}=-\left[\gamma -{\frac {\ln 2}{2}}\right]\ln 2+\,i\!\int _{0}^{\infty }\!{\frac {\mathrm {d} x}{e^{\pi x}+1}}\left\{{\frac {\ln(1-ix)}{1-ix}}-{\frac {\ln(1+ix)}{1+ix}}\right\}\,\\[6mm]\displaystyle \gamma _{1}=-\gamma ^{2}-\int _{0}^{\infty }\left[{\frac {1}{1-e^{-x}}}-{\frac {1}{x}}\right]e^{-x}\ln x\,\mathrm {d} x\!\,.\end{array}}}
Znano vrsto, ki vsebuje celi del logaritma , je podal Hardy leta 1912:[10]
γ
1
=
ln
2
2
∑
k
=
2
∞
(
−
1
)
k
k
⌊
lb
k
⌋
⋅
(
2
lb
k
−
⌊
lb
(
2
k
)
⌋
)
.
{\displaystyle \gamma _{1}\,=\,{\frac {\ln 2}{2}}\sum _{k=2}^{\infty }{\frac {(-1)^{k}}{k}}\,\lfloor \operatorname {lb} k\rfloor \cdot {\big (}2\operatorname {lb} k-\lfloor \operatorname {lb} (2k)\rfloor {\big )}\!\,.}
Tu je
lb
{\displaystyle \operatorname {lb} \,}
dvojiški logaritem .
Israilov je podal delno konvergentno vrsto z Bernoullijevimi števili
B
2
k
{\displaystyle B_{2k}\,}
:[11]
γ
m
=
∑
k
=
1
n
ln
m
k
k
−
ln
m
+
1
n
m
+
1
−
ln
m
n
2
n
−
∑
k
=
1
N
−
1
B
2
k
(
2
k
)
!
[
ln
m
x
x
]
x
=
n
(
2
k
−
1
)
−
θ
⋅
B
2
N
(
2
N
)
!
[
ln
m
x
x
]
x
=
n
(
2
N
−
1
)
,
(
0
<
θ
<
1
)
.
{\displaystyle \gamma _{m}\,=\,\sum _{k=1}^{n}{\frac {\,\ln ^{m}\!k\,}{k}}-{\frac {\,\ln ^{m+1}\!n\,}{m+1}}-{\frac {\,\ln ^{m}\!n\,}{2n}}-\sum _{k=1}^{N-1}{\frac {\,B_{2k}\,}{(2k)!}}\left[{\frac {\ln ^{m}\!x}{x}}\right]_{x=n}^{(2k-1)}-\theta \cdot {\frac {\,B_{2N}\,}{(2N)!}}\left[{\frac {\ln ^{m}\!x}{x}}\right]_{x=n}^{(2N-1)}\,,\qquad (0<\theta <1)\!\,.}
Oloa in Tauraso sta pokazala, da vrsta s harmoničnimi števili
H
n
{\displaystyle H_{n}\,}
lahko vodi do Stieltjesovih konstant:[12]
∑
n
=
1
∞
H
n
−
(
γ
+
ln
n
)
n
=
−
γ
1
−
1
2
γ
2
+
1
12
π
2
∑
n
=
1
∞
H
n
2
−
(
γ
+
ln
n
)
2
n
=
−
γ
2
−
2
γ
γ
1
−
2
3
γ
3
+
5
3
ζ
(
3
)
.
{\displaystyle {\begin{array}{l}\displaystyle \sum _{n=1}^{\infty }{\frac {\,H_{n}-(\gamma +\ln n)\,}{n}}\,=\,\,-\gamma _{1}-{\frac {1}{2}}\gamma ^{2}+{\frac {1}{12}}\pi ^{2}\\[6mm]\displaystyle \sum _{n=1}^{\infty }{\frac {\,H_{n}^{2}-(\gamma +\ln n)^{2}\,}{n}}\,=\,\,-\gamma _{2}-2\gamma \gamma _{1}-{\frac {2}{3}}\gamma ^{3}+{\frac {5}{3}}\zeta (3)\!\,.\end{array}}}
Blagouchine je našel počasi konvergentno vrsto, ki vsebuje nepredznačena Stirlingova števila prve vrste
[
⋅
⋅
]
{\displaystyle \left[{\cdot \atop \cdot }\right]\,}
:[8]
γ
m
=
1
2
δ
m
,
0
+
(
−
1
)
m
m
!
π
∑
n
=
1
∞
1
n
⋅
n
!
∑
k
=
0
⌊
1
2
n
⌋
(
−
1
)
k
⋅
[
2
k
+
2
m
+
1
]
⋅
[
n
2
k
+
1
]
(
2
π
)
2
k
+
1
,
(
m
=
0
,
1
,
2
,
…
)
,
{\displaystyle \gamma _{m}\,=\,{\frac {1}{2}}\delta _{m,0}+{\frac {\,(-1)^{m}m!\,}{\pi }}\sum _{n=1}^{\infty }{\frac {1}{\,n\cdot n!\,}}\sum _{k=0}^{\lfloor \!{\frac {1}{2}}n\!\rfloor }{\frac {\,(-1)^{k}\cdot \left[{2k+2 \atop m+1}\right]\cdot \left[{n \atop 2k+1}\right]\,}{\,(2\pi )^{2k+1}\,}}\,,\qquad (m=0,1,2,\ldots )\!\,,}
kot tudi delno konvergentno vrsto s samimi racionalnimi členi:
γ
m
=
1
2
δ
m
,
0
+
(
−
1
)
m
m
!
⋅
∑
k
=
1
N
[
2
k
m
+
1
]
⋅
B
2
k
(
2
k
)
!
+
θ
⋅
(
−
1
)
m
m
!
⋅
[
2
N
+
2
m
+
1
]
⋅
B
2
N
+
2
(
2
N
+
2
)
!
,
(
0
<
θ
<
1
,
m
=
0
,
1
,
2
,
…
)
.
{\displaystyle \gamma _{m}\,=\,{\frac {1}{2}}\delta _{m,0}+(-1)^{m}m!\cdot \!\sum _{k=1}^{N}{\frac {\,\left[{2k \atop m+1}\right]\cdot B_{2k}\,}{(2k)!}}\,+\,\theta \cdot {\frac {\,(-1)^{m}m!\!\cdot \left[{2N+2 \atop m+1}\right]\cdot B_{2N+2}\,}{(2N+2)!}}\,,\qquad (0<\theta <1,m=0,1,2,\ldots )\!\,.}
Več drugih vrst je danih v Coffeyjevemu delu.[4] [5]
Za Stieltjesove konstante velja meja:
|
γ
n
|
≤
{
2
(
n
−
1
)
!
π
n
;
n
=
1
,
3
,
5
,
…
4
(
n
−
1
)
!
π
n
;
n
=
2
,
4
,
6
,
…
,
{\displaystyle {\big |}\gamma _{n}{\big |}\,\leq \,{\begin{cases}\displaystyle {\frac {2\,(n-1)!}{\pi ^{n}}}\,;&n=1,3,5,\ldots \\[3mm]\displaystyle {\frac {4\,(n-1)!}{\pi ^{n}}}\,;&n=2,4,6,\ldots \!\,,\end{cases}}}
ki jo je podal Berndt leta 1972.[13] Boljše meje so našli Lavrik, Israilov, Matsuoka, Nan-You, Williams, Knessl, Coffey, Adell, Saad-Eddin, Fekih-Ahmed in Blagouchine.[b] Eno od najboljših ocen z elementarnimi funkcijami je podal Matsuoka leta 1985:[14]
|
γ
n
|
<
10
−
4
e
n
ln
ln
n
,
(
n
≥
5
)
.
{\displaystyle |\gamma _{n}|<10^{-4}e^{n\ln \ln n}\,,\qquad (n\geq 5)\!\,.}
Dokaj točne ocene z neelementarnimi funkcijami so podali Knessl, Coffey[15] in Fekih-Ahmed.[16] Knessl in Coffey sta na primer dala naslednjo formulo, ki relativno dobro aproksimira Stieltjesove konstante za velike
n
{\displaystyle n\,}
.[15] Če je
v
{\displaystyle v\,}
enolična rešitev enačbe:
2
π
exp
(
v
tg
v
)
=
n
cos
v
v
,
{\displaystyle 2\pi \exp(v\operatorname {tg} \,v)=n{\frac {\cos v}{v}}\!\,,}
z
0
<
v
<
π
/
2
{\displaystyle 0<v<\pi /2\,}
, in, če je
u
=
v
tg
v
{\displaystyle u=v\operatorname {tg} \,v\,}
, potem velja:
γ
n
∼
B
n
e
n
A
cos
(
a
n
+
b
)
,
{\displaystyle \gamma _{n}\sim {\frac {B}{\sqrt {n}}}e^{nA}\cos(an+b)\!\,,}
kjer je:
A
=
1
2
ln
(
u
2
+
v
2
)
−
u
u
2
+
v
2
,
{\displaystyle A={\frac {1}{2}}\ln(u^{2}+v^{2})-{\frac {u}{u^{2}+v^{2}}}\!\,,}
B
=
2
2
π
u
2
+
v
2
[
(
u
+
1
)
2
+
v
2
]
1
/
4
,
{\displaystyle B={\frac {2{\sqrt {2\pi }}{\sqrt {u^{2}+v^{2}}}}{[(u+1)^{2}+v^{2}]^{1/4}}}\!\,,}
a
=
tg
−
1
(
v
u
)
+
v
u
2
+
v
2
,
{\displaystyle a=\operatorname {tg} ^{-1}\left({\frac {v}{u}}\right)+{\frac {v}{u^{2}+v^{2}}}\!\,,}
b
=
tg
−
1
(
v
u
)
−
1
2
(
v
u
+
1
)
.
{\displaystyle b=\operatorname {tg} ^{-1}\left({\frac {v}{u}}\right)-{\frac {1}{2}}\left({\frac {v}{u+1}}\right)\!\,.}
Vse do
n
=
100000
{\displaystyle n=100000\,}
Knessl-Coffeyjev približek trenutno predvideva predznak
γ
n
{\displaystyle \gamma _{n}\,}
z eno izjemo za
n
=
137
{\displaystyle n=137\,}
.[15]
Prve desetiške vrednosti Stieltjesovih konstant podaja razpredelnica:
n
{\displaystyle n\,}
desetiške vrednosti
γ
n
{\displaystyle \gamma _{n}\,}
OEIS
0
+0,5772156649015328606065120900824024310421593359
A001620
1
−0,0728158454836767248605863758749013191377363383
A082633
2
−0,0096903631928723184845303860352125293590658061
A086279
3
+0,0020538344203033458661600465427533842857158044
A086280
4
+0,0023253700654673000574681701775260680009044694
A086281
5
+0,0007933238173010627017533348774444448307315394
A086282
6
−0,0002387693454301996098724218419080042777837151
A183141
7
−0,0005272895670577510460740975054788582819962534
A183167
8
−0,0003521233538030395096020521650012087417291805
A183206
9
−0,0000343947744180880481779146237982273906207895
A184853
10
+0,0002053328149090647946837222892370653029598537
A184854
100
−4,2534015717080269623144385197278358247028931053 · 1017
1000
−1,5709538442047449345494023425120825242380299554 · 10486
10000
−2,2104970567221060862971082857536501900234397174 · 106883
100000
+1,9919273063125410956582272431568589205211659777 · 1083432
Za velike
n
{\displaystyle n\,}
absolutne vrednosti Stieltjesovih konstant naraščajo hitro, predznak pa se spreminja v zapletenem vzorcu.
Dodatne informacije o numeričnem določevanju Stieltjesovih konstant se lahko najde v delu avtorjev: Keiper ,[17] Kreminski,[18] Plouffe [19] in Johansson.[20] Johansson je podal vrednosti Stieltjesovih konstant do
n
=
100000
{\displaystyle n=100000\,}
, vsaka točna na več kot 10000 števk. Številske vrednosti se lahko dobijo v podatkovni bazi LMFDB .[21]
Posplošene Stieltjesove konstante [ uredi | uredi kodo ]
Bolj splošno se lahko definirajo Stieltjesove konstante
γ
n
(
a
)
{\displaystyle \gamma _{n}(a)\,}
, ki se pojavljajo v Laurentovi vrsti za Hurwitzevo funkcijo ζ :
ζ
(
s
,
a
)
=
1
s
−
1
+
∑
n
=
0
∞
(
−
1
)
n
n
!
γ
n
(
a
)
(
s
−
1
)
n
.
{\displaystyle \zeta (s,a)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}(a)\;(s-1)^{n}\!\,.}
Tu je
a
{\displaystyle a\,}
kompleksno število z
ℜ
(
a
)
>
0
{\displaystyle \Re (a)>0\,}
. Ker je Hurwitzeva funkcija ζ posplošitev Riemannove funkcije ζ, velja
γ
n
(
1
)
=
γ
n
{\displaystyle \gamma _{n}(1)=\gamma _{n}\,}
. Ničta konstanta je preprosto funkcija digama
γ
0
(
a
)
=
−
ϝ
(
a
)
{\displaystyle \gamma _{0}(a)=-\digamma (a)\,}
.[22] Za druge konstante ni znana razčlenitev na elementarne ali klasične funkcije iz analize. Ne glede na to obstaja več izrazov zanje. Na primer naslednji asimptotični izraz:
γ
n
(
a
)
=
lim
m
→
∞
{
∑
k
=
0
m
ln
n
(
k
+
a
)
k
+
a
−
ln
n
+
1
(
m
+
a
)
n
+
1
}
,
n
=
0
,
1
,
2
,
…
a
≠
0
,
−
1
,
−
2
,
…
,
{\displaystyle \gamma _{n}(a)\,=\,\lim _{m\to \infty }\left\{\sum _{k=0}^{m}{\frac {\ln ^{n}(k+a)}{k+a}}-{\frac {\ln ^{n+1}(m+a)}{n+1}}\right\}\,,\qquad \;{\begin{array}{l}n=0,1,2,\ldots \,\\[1mm]a\neq 0,-1,-2,\ldots \!\,,\end{array}}}
ki sta jo podala Berndt in Wilton. Analogon Jensen-Franelove formule za posplošeno Stieltjesovo konstanto je Hermitova formula:[7]
γ
n
(
a
)
=
[
1
2
a
−
ln
a
n
+
1
]
ln
n
a
−
i
∫
0
∞
d
x
e
2
π
x
−
1
{
ln
n
(
a
−
i
x
)
a
−
i
x
−
ln
n
(
a
+
i
x
)
a
+
i
x
}
,
n
=
0
,
1
,
2
,
…
a
≠
0
,
−
1
,
−
2
,
…
{\displaystyle \gamma _{n}(a)\,=\,\left[{\frac {1}{2a}}-{\frac {\ln {a}}{n+1}}\right]\ln ^{n}\!{a}-i\!\int _{0}^{\infty }\!{\frac {\mathrm {d} x}{e^{2\pi x}-1}}\left\{{\frac {\ln ^{n}(a-ix)}{a-ix}}-{\frac {\ln ^{n}(a+ix)}{a+ix}}\right\}\,,\qquad \;{\begin{array}{l}n=0,1,2,\ldots \,\\[1mm]a\neq 0,-1,-2,\ldots \end{array}}}
Za posplošene Stieltjesove konstante velja naslednja rekurenčna zveza:
γ
n
(
a
+
1
)
=
γ
n
(
a
)
−
ln
n
a
a
,
n
=
0
,
1
,
2
,
…
a
≠
0
,
−
1
,
−
2
,
…
,
{\displaystyle \gamma _{n}(a+1)\,=\,\gamma _{n}(a)-{\frac {\,\ln ^{n}\!a\,}{a}}\,,\qquad \;{\begin{array}{l}n=0,1,2,\ldots \,\\[1mm]a\neq 0,-1,-2,\ldots \!\,,\end{array}}}
kakor tudi multiplikacijski izrek:
∑
l
=
0
n
−
1
γ
p
(
a
+
l
n
)
=
(
−
1
)
p
n
[
ln
n
p
+
1
−
ϝ
(
a
n
)
]
ln
p
n
+
n
∑
r
=
0
p
−
1
(
−
1
)
r
(
p
r
)
γ
p
−
r
(
a
n
)
⋅
ln
r
n
,
(
n
=
2
,
3
,
4
,
…
)
,
{\displaystyle \sum _{l=0}^{n-1}\gamma _{p}\!\left(\!a+{\frac {l}{\,n\,}}\right)=\,(-1)^{p}n\!\left[{\frac {\ln n}{\,p+1\,}}-\digamma (an)\right]\!\ln ^{p}\!n\,+\,n\sum _{r=0}^{p-1}(-1)^{r}{\binom {p}{r}}\gamma _{p-r}(an)\cdot \ln ^{r}\!{n}\,,\qquad (n=2,3,4,\ldots )\!\,,}
kjer
(
p
r
)
{\displaystyle {\binom {p}{r}}}
označuje binomski koeficient .[23] [24] :101–102
Prva posplošena Stieltjesova konstanta [ uredi | uredi kodo ]
Prva posplošena Stieltjesova konstanta ima več pomembnih značilnosti.
Malmstenova enakost (refleksijska formula za prve posplošene Stieltjesove konstante): refleksijska formula za prvo posplošeno Stieltjesovo konstanto ima obliko:
γ
1
(
m
n
)
−
γ
1
(
1
−
m
n
)
=
2
π
∑
l
=
1
n
−
1
sin
2
π
m
l
n
⋅
ln
Γ
(
l
n
)
−
π
(
γ
+
ln
2
π
n
)
cot
m
π
n
,
{\displaystyle \gamma _{1}{\biggl (}{\frac {m}{n}}{\biggr )}-\gamma _{1}{\biggl (}1-{\frac {m}{n}}{\biggr )}=2\pi \sum _{l=1}^{n-1}\sin {\frac {2\pi ml}{n}}\cdot \ln \Gamma {\biggl (}{\frac {l}{n}}{\biggr )}-\pi (\gamma +\ln 2\pi n)\cot {\frac {m\pi }{n}}\!\,,}
kjer sta
m
{\displaystyle m\,}
in
n
{\displaystyle n\,}
takšni pozitivni celi števii, da velja
m
<
n
{\displaystyle m<n\,}
,
Γ
{\displaystyle \Gamma \,}
pa je funkcija Γ . Formulo so dolgo časa pripisovali Almkvistu in Meurmanu, ki sta jo izpeljala v 1990-ih.[25] Vendar je nedavno Blagouchine odkril, da je to enakost, sicer v malo drugačni obliki, našel Malmsten leta 1846.[7] [26]
Izrek o racionalnih argumentih: prva posplošena Stieltjesova konstanta z racionalnim argumentom se lahko izračuna iz delno sklenjene oblike s formulo:[7] [22]
γ
1
(
r
m
)
=
γ
1
+
γ
2
+
γ
ln
2
π
m
+
ln
2
π
⋅
ln
m
+
1
2
ln
2
m
+
(
γ
+
ln
2
π
m
)
⋅
ϝ
(
r
m
)
+
π
∑
l
=
1
m
−
1
sin
2
π
r
l
m
⋅
ln
Γ
(
l
m
)
+
∑
l
=
1
m
−
1
cos
2
π
r
l
m
⋅
ζ
″
(
0
,
l
m
)
,
(
r
=
1
,
2
,
3
,
…
,
m
−
1
)
.
{\displaystyle {\begin{array}{ll}\displaystyle \gamma _{1}{\biggl (}{\frac {r}{m}}{\biggr )}=&\displaystyle \gamma _{1}+\gamma ^{2}+\gamma \ln 2\pi m+\ln 2\pi \cdot \ln {m}+{\frac {1}{2}}\ln ^{2}\!{m}+(\gamma +\ln 2\pi m)\cdot \digamma \!\left(\!{\frac {r}{m}}\!\right)\\[5mm]\displaystyle &\displaystyle \qquad +\pi \sum _{l=1}^{m-1}\sin {\frac {2\pi rl}{m}}\cdot \ln \Gamma {\biggl (}{\frac {l}{m}}{\biggr )}+\sum _{l=1}^{m-1}\cos {\frac {2\pi rl}{m}}\cdot \zeta ''\!\left(\!0,\,{\frac {l}{m}}\!\right)\end{array}}\,,\qquad \quad (r=1,2,3,\ldots ,m-1)\!\,.}
Alternativni dokaz je kasneje predložil Coffey.[27]
Končne vsote: za prve posplošene Stieltjesove konstante obstaje veliko sumacijskih formul. Na primer:[c]
∑
r
=
0
m
−
1
γ
1
(
a
+
r
m
)
=
m
ln
m
⋅
ϝ
(
a
m
)
−
m
2
ln
2
m
+
m
γ
1
(
a
m
)
,
(
a
∈
C
)
∑
r
=
1
m
−
1
γ
1
(
r
m
)
=
(
m
−
1
)
γ
1
−
m
γ
ln
m
−
m
2
ln
2
m
∑
r
=
1
2
m
−
1
(
−
1
)
r
γ
1
(
r
2
m
)
=
−
γ
1
+
m
(
2
γ
+
ln
2
+
2
ln
m
)
ln
2
∑
r
=
0
2
m
−
1
(
−
1
)
r
γ
1
(
2
r
+
1
4
m
)
=
m
{
4
π
ln
Γ
(
1
4
)
−
π
(
4
ln
2
+
3
ln
π
+
ln
m
+
γ
)
}
∑
r
=
1
m
−
1
γ
1
(
r
m
)
⋅
cos
2
π
r
k
m
=
−
γ
1
+
m
(
γ
+
ln
2
π
m
)
ln
(
2
sin
k
π
m
)
+
m
2
{
ζ
″
(
0
,
k
m
)
+
ζ
″
(
0
,
1
−
k
m
)
}
,
(
k
=
1
,
2
,
…
,
m
−
1
)
∑
r
=
1
m
−
1
γ
1
(
r
m
)
⋅
sin
2
π
r
k
m
=
π
2
(
γ
+
ln
2
π
m
)
(
2
k
−
m
)
−
π
m
2
{
ln
π
−
ln
sin
k
π
m
}
+
m
π
ln
Γ
(
k
m
)
,
(
k
=
1
,
2
,
…
,
m
−
1
)
∑
r
=
1
m
−
1
γ
1
(
r
m
)
⋅
cot
π
r
m
=
π
6
{
(
1
−
m
)
(
m
−
2
)
γ
+
2
(
m
2
−
1
)
ln
2
π
−
(
m
2
+
2
)
ln
m
}
−
2
π
∑
l
=
1
m
−
1
l
⋅
ln
Γ
(
l
m
)
∑
r
=
1
m
−
1
r
m
⋅
γ
1
(
r
m
)
=
1
2
{
(
m
−
1
)
γ
1
−
m
γ
ln
m
−
m
2
ln
2
m
}
−
π
2
m
(
γ
+
ln
2
π
m
)
∑
l
=
1
m
−
1
l
⋅
cot
π
l
m
−
π
2
∑
l
=
1
m
−
1
cot
π
l
m
⋅
ln
Γ
(
l
m
)
.
{\displaystyle {\begin{array}{ll}\displaystyle \sum _{r=0}^{m-1}\gamma _{1}\!\left(\!a+{\frac {r}{\,m\,}}\right)=\,m\ln {m}\cdot \digamma (am)-{\frac {m}{2}}\ln ^{2}\!m+m\gamma _{1}(am)\,,\qquad (a\in \mathbb {C} )\\[6mm]\displaystyle \sum _{r=1}^{m-1}\gamma _{1}\!\left(\!{\frac {r}{\,m\,}}\right)=\,(m-1)\gamma _{1}-m\gamma \ln {m}-{\frac {m}{2}}\ln ^{2}\!m\\[6mm]\displaystyle \sum _{r=1}^{2m-1}(-1)^{r}\gamma _{1}{\biggl (}\!{\frac {r}{2m}}\!{\biggr )}\,=\,-\gamma _{1}+m(2\gamma +\ln 2+2\ln m)\ln 2\\[6mm]\displaystyle \sum _{r=0}^{2m-1}(-1)^{r}\gamma _{1}{\biggl (}\!{\frac {2r+1}{4m}}\!{\biggr )}\,=\,m\left\{4\pi \ln \Gamma {\biggl (}{\frac {1}{4}}{\biggr )}-\pi {\big (}4\ln 2+3\ln \pi +\ln m+\gamma {\big )}\!\right\}\\[6mm]\displaystyle \sum _{r=1}^{m-1}\gamma _{1}{\biggl (}\!{\frac {r}{m}}\!{\biggr )}\!\cdot \cos {\dfrac {2\pi rk}{m}}\,=\,-\gamma _{1}+m(\gamma +\ln 2\pi m)\ln \!\left(\!2\sin {\frac {\,k\pi \,}{m}}\!\right)+{\frac {m}{2}}\left\{\zeta ''\!\left(\!0,\,{\frac {k}{m}}\!\right)+\,\zeta ''\!\left(\!0,\,1-{\frac {k}{m}}\!\right)\!\right\}\,,\qquad (k=1,2,\ldots ,m-1)\\[6mm]\displaystyle \sum _{r=1}^{m-1}\gamma _{1}{\biggl (}\!{\frac {r}{m}}\!{\biggr )}\!\cdot \sin {\dfrac {2\pi rk}{m}}\,=\,{\frac {\pi }{2}}(\gamma +\ln 2\pi m)(2k-m)-{\frac {\pi m}{2}}\left\{\ln \pi -\ln \sin {\frac {k\pi }{m}}\right\}+m\pi \ln \Gamma {\biggl (}{\frac {k}{m}}{\biggr )}\,,\qquad (k=1,2,\ldots ,m-1)\\[6mm]\displaystyle \sum _{r=1}^{m-1}\gamma _{1}{\biggl (}\!{\frac {r}{m}}\!{\biggr )}\cdot \cot {\frac {\pi r}{m}}=\,\displaystyle {\frac {\pi }{6}}{\Big \{}\!(1-m)(m-2)\gamma +2(m^{2}-1)\ln 2\pi -(m^{2}+2)\ln {m}{\Big \}}-2\pi \!\sum _{l=1}^{m-1}l\!\cdot \!\ln \Gamma \!\left(\!{\frac {l}{m}}\!\right)\\[6mm]\displaystyle \sum _{r=1}^{m-1}{\frac {r}{m}}\cdot \gamma _{1}{\biggl (}\!{\frac {r}{m}}\!{\biggr )}=\,{\frac {1}{2}}\left\{\!(m-1)\gamma _{1}-m\gamma \ln {m}-{\frac {m}{2}}\ln ^{2}\!{m}\!\right\}-{\frac {\pi }{2m}}(\gamma +\ln 2\pi m)\!\sum _{l=1}^{m-1}l\!\cdot \!\cot {\frac {\pi l}{m}}-{\frac {\pi }{2}}\!\sum _{l=1}^{m-1}\cot {\frac {\pi l}{m}}\cdot \ln \Gamma {\biggl (}\!{\frac {l}{m}}\!{\biggr )}\!\,.\end{array}}}
Nekatere posebne vrednosti: nekatere posebne vrednosti prve Stieltjesove konstante z racionalnimi argumenti se lahko zreducirajo na funkcijo Γ, prvo Stieltjesovo konstanto
γ
1
{\displaystyle \gamma _{1}\,}
in elementarne funkcije. Na primer:
γ
1
(
1
2
)
=
−
2
γ
ln
2
−
ln
2
2
+
γ
1
=
−
1
,
353459680804
…
,
{\displaystyle \gamma _{1}\!\left(\!{\frac {1}{\,2\,}}\!\right)=-2\gamma \ln 2-\ln ^{2}\!2+\gamma _{1}\,=\,-1,353459680804\ldots \!\,,}
(OEIS A254327 ),
Vrednosti prvih posplošenih Stieltjesovih konstant v točkah 1/4, 3/4 in 1/3 sta prva neodvisno izračunala Connon[28] in Blagouchine:[24]
γ
1
(
1
4
)
=
2
π
ln
Γ
(
1
4
)
−
3
π
2
ln
π
−
7
2
ln
2
2
−
(
3
γ
+
2
π
)
ln
2
−
γ
π
2
+
γ
1
=
−
5
,
518076350199
…
,
{\displaystyle \displaystyle \gamma _{1}\!\left(\!{\frac {1}{\,4\,}}\!\right)=\,2\pi \ln \Gamma \!\left(\!{\frac {1}{\,4\,}}\!\right)-{\frac {3\pi }{2}}\ln \pi -{\frac {7}{2}}\ln ^{2}\!2-(3\gamma +2\pi )\ln 2-{\frac {\gamma \pi }{2}}+\gamma _{1}\,=\,-5,518076350199\ldots \!\,,}
(OEIS A254347 ),
γ
1
(
3
4
)
=
−
2
π
ln
Γ
(
1
4
)
+
3
π
2
ln
π
−
7
2
ln
2
2
−
(
3
γ
−
2
π
)
ln
2
+
γ
π
2
+
γ
1
=
−
0
,
391298902404
…
,
{\displaystyle \displaystyle \gamma _{1}\!\left(\!{\frac {3}{\,4\,}}\!\right)=\,-2\pi \ln \Gamma \!\left(\!{\frac {1}{\,4\,}}\!\right)+{\frac {3\pi }{2}}\ln \pi -{\frac {7}{2}}\ln ^{2}\!2-(3\gamma -2\pi )\ln 2+{\frac {\gamma \pi }{2}}+\gamma _{1}\,=\,-0,391298902404\ldots \!\,,}
(OEIS A254348 ),
γ
1
(
1
3
)
=
−
3
γ
2
ln
3
−
3
4
ln
2
3
+
π
4
3
{
ln
3
−
8
ln
2
π
−
2
γ
+
12
ln
Γ
(
1
3
)
}
+
γ
1
=
−
3
,
259557515917
…
,
{\displaystyle \displaystyle \gamma _{1}\!\left(\!{\frac {1}{\,3\,}}\!\right)=\,-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}\ln ^{2}\!3+{\frac {\pi }{4{\sqrt {3\,}}}}\left\{\ln 3-8\ln 2\pi -2\gamma +12\ln \Gamma \!\left(\!{\frac {1}{\,3\,}}\!\right)\!\right\}+\,\gamma _{1}\,=\,-3,259557515917\ldots \!\,,}
(OEIS A254331 ).
Vrednosti v točkah 2/3, 1/6 in 5/6 je izračunal Blagouchine:[24]
γ
1
(
2
3
)
=
−
3
γ
2
ln
3
−
3
4
ln
2
3
−
π
4
3
{
ln
3
−
8
ln
2
π
−
2
γ
+
12
ln
Γ
(
1
3
)
}
+
γ
1
=
−
0
,
5989062842859
…
,
{\displaystyle \displaystyle \gamma _{1}\!\left(\!{\frac {2}{\,3\,}}\!\right)=\,-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}\ln ^{2}\!3-{\frac {\pi }{4{\sqrt {3\,}}}}\left\{\ln 3-8\ln 2\pi -2\gamma +12\ln \Gamma \!\left(\!{\frac {1}{\,3\,}}\!\right)\!\right\}+\,\gamma _{1}\,=\,-0,5989062842859\ldots \!\,,}
(OEIS A254345 ),
γ
1
(
1
6
)
=
−
3
γ
2
ln
3
−
3
4
ln
2
3
−
ln
2
2
−
(
3
ln
3
+
2
γ
)
ln
2
+
3
π
3
2
ln
Γ
(
1
6
)
−
π
2
3
{
3
ln
3
+
11
ln
2
+
15
2
ln
π
+
3
γ
}
+
γ
1
=
−
10
,
742582529547
…
,
{\displaystyle {\begin{aligned}\displaystyle \gamma _{1}\!\left(\!{\frac {1}{\,6\,}}\!\right)=&-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}\ln ^{2}\!3-\ln ^{2}\!2-(3\ln 3+2\gamma )\ln 2+{\frac {3\pi {\sqrt {3\,}}}{2}}\ln \Gamma \!\left(\!{\frac {1}{\,6\,}}\!\right)\\\displaystyle &-{\frac {\pi }{2{\sqrt {3\,}}}}\left\{3\ln 3+11\ln 2+{\frac {15}{2}}\ln \pi +3\gamma \right\}+\,\gamma _{1}\,=\,-10,742582529547\ldots \!\,,\end{aligned}}}
(OEIS A254349 ),
γ
1
(
5
6
)
=
−
3
γ
2
ln
3
−
3
4
ln
2
3
−
ln
2
2
−
(
3
ln
3
+
2
γ
)
ln
2
−
3
π
3
2
ln
Γ
(
1
6
)
+
π
2
3
{
3
ln
3
+
11
ln
2
+
15
2
ln
π
+
3
γ
}
+
γ
1
=
−
0
,
246169003811
…
,
{\displaystyle {\begin{aligned}\displaystyle \gamma _{1}\!\left(\!{\frac {5}{\,6\,}}\!\right)=&-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}\ln ^{2}\!3-\ln ^{2}\!2-(3\ln 3+2\gamma )\ln 2-{\frac {3\pi {\sqrt {3\,}}}{2}}\ln \Gamma \!\left(\!{\frac {1}{\,6\,}}\!\right)\\\displaystyle &+{\frac {\pi }{2{\sqrt {3\,}}}}\left\{3\ln 3+11\ln 2+{\frac {15}{2}}\ln \pi +3\gamma \right\}+\,\gamma _{1}\,=\,-0,246169003811\ldots \!\,,\end{aligned}}}
(OEIS A254350 ),
Podal je tudi vrednosti v točkah 1/5, 1/8 in 1/12:
γ
1
(
1
5
)
=
γ
1
+
5
2
{
ζ
″
(
0
,
1
5
)
+
ζ
″
(
0
,
4
5
)
}
+
π
10
+
2
5
2
ln
Γ
(
1
5
)
+
π
10
−
2
5
2
ln
Γ
(
2
5
)
+
{
5
2
ln
2
−
5
2
ln
(
1
+
5
)
−
5
4
ln
5
−
π
25
+
10
5
10
}
⋅
γ
−
5
2
{
ln
2
+
ln
5
+
ln
π
+
π
25
−
10
5
10
}
⋅
ln
(
1
+
5
)
+
5
2
ln
2
2
+
5
(
1
−
5
)
8
ln
2
5
+
3
5
4
ln
2
⋅
ln
5
+
5
2
ln
2
⋅
ln
π
+
5
4
ln
5
⋅
ln
π
−
π
(
2
25
+
10
5
+
5
25
+
2
5
)
20
ln
2
−
π
(
4
25
+
10
5
−
5
5
+
2
5
)
40
ln
5
−
π
(
5
5
+
2
5
+
25
+
10
5
)
10
ln
π
=
−
8
,
030205511035
…
,
{\displaystyle {\begin{aligned}\displaystyle \gamma _{1}{\biggl (}\!{\frac {1}{5}}\!{\biggr )}=&\displaystyle \,\,\,\gamma _{1}+{\frac {\sqrt {5}}{2}}\!\left\{\zeta ''\!\left(\!0,\,{\frac {1}{5}}\!\right)+\zeta ''\!\left(\!0,\,{\frac {4}{5}}\!\right)\!\right\}+{\frac {\pi {\sqrt {10+2{\sqrt {5}}}}}{2}}\ln \Gamma {\biggl (}\!{\frac {1}{5}}\!{\biggr )}\\[5mm]&\displaystyle +{\frac {\pi {\sqrt {10-2{\sqrt {5}}}}}{2}}\ln \Gamma {\biggl (}\!{\frac {2}{5}}\!{\biggr )}+\left\{\!{\frac {\sqrt {5}}{2}}\ln {2}-{\frac {\sqrt {5}}{2}}\ln \!{\big (}1+{\sqrt {5}}{\big )}-{\frac {5}{4}}\ln 5-{\frac {\pi {\sqrt {25+10{\sqrt {5}}}}}{10}}\right\}\!\cdot \gamma \\[5mm]&\displaystyle -{\frac {\sqrt {5}}{2}}\left\{\ln 2+\ln 5+\ln \pi +{\frac {\pi {\sqrt {25-10{\sqrt {5}}}}}{10}}\right\}\!\cdot \ln \!{\big (}1+{\sqrt {5}})+{\frac {\sqrt {5}}{2}}\ln ^{2}\!2+{\frac {{\sqrt {5}}{\big (}1-{\sqrt {5}}{\big )}}{8}}\ln ^{2}\!5\\[5mm]&\displaystyle +{\frac {3{\sqrt {5}}}{4}}\ln 2\cdot \ln 5+{\frac {\sqrt {5}}{2}}\ln 2\cdot \ln \pi +{\frac {\sqrt {5}}{4}}\ln 5\cdot \ln \pi -{\frac {\pi {\big (}2{\sqrt {25+10{\sqrt {5}}}}+5{\sqrt {25+2{\sqrt {5}}}}{\big )}}{20}}\ln 2\\[5mm]&\displaystyle -{\frac {\pi {\big (}4{\sqrt {25+10{\sqrt {5}}}}-5{\sqrt {5+2{\sqrt {5}}}}{\big )}}{40}}\ln 5-{\frac {\pi {\big (}5{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {25+10{\sqrt {5}}}}{\big )}}{10}}\ln \pi \\[5mm]&\displaystyle =-8,030205511035\ldots \!\,,\end{aligned}}}
(OEIS A251866 ),
γ
1
(
1
8
)
=
γ
1
+
2
{
ζ
″
(
0
,
1
8
)
+
ζ
″
(
0
,
7
8
)
}
+
2
π
2
ln
Γ
(
1
8
)
−
π
2
(
1
−
2
)
ln
Γ
(
1
4
)
−
{
1
+
2
2
π
+
4
ln
2
+
2
ln
(
1
+
2
)
}
⋅
γ
−
1
2
(
π
+
8
ln
2
+
2
ln
π
)
⋅
ln
(
1
+
2
)
−
7
(
4
−
2
)
4
ln
2
2
+
1
2
ln
2
⋅
ln
π
−
π
(
10
+
11
2
)
4
ln
2
−
π
(
3
+
2
2
)
2
ln
π
=
−
16
,
641719763609
…
{\displaystyle {\begin{aligned}\displaystyle \gamma _{1}{\biggl (}\!{\frac {1}{8}}\!{\biggr )}=&\displaystyle \,\,\,\gamma _{1}+{\sqrt {2}}\left\{\zeta ''\!\left(\!0,\,{\frac {1}{8}}\!\right)+\zeta ''\!\left(\!0,\,{\frac {7}{8}}\right)\!\right\}+2\pi {\sqrt {2}}\ln \Gamma {\biggl (}\!{\frac {1}{8}}\!{\biggr )}-\pi {\sqrt {2}}{\big (}1-{\sqrt {2}}{\big )}\ln \Gamma {\biggl (}\!{\frac {1}{4}}\!{\biggr )}\\[5mm]&\displaystyle -\left\{\!{\frac {1+{\sqrt {2}}}{2}}\pi +4\ln {2}+{\sqrt {2}}\ln \!{\big (}1+{\sqrt {2}}{\big )}\!\right\}\!\cdot \gamma -{\frac {1}{\sqrt {2}}}{\big (}\pi +8\ln 2+2\ln \pi {\big )}\!\cdot \ln \!{\big (}1+{\sqrt {2}})\\[5mm]&\displaystyle -{\frac {7{\big (}4-{\sqrt {2}}{\big )}}{4}}\ln ^{2}\!2+{\frac {1}{\sqrt {2}}}\ln 2\cdot \ln \pi -{\frac {\pi {\big (}10+11{\sqrt {2}}{\big )}}{4}}\ln 2-{\frac {\pi {\big (}3+2{\sqrt {2}}{\big )}}{2}}\ln \pi \\[5mm]&\displaystyle =-16,641719763609\ldots \end{aligned}}}
(OEIS A255188 ),
γ
1
(
1
12
)
=
γ
1
+
3
{
ζ
″
(
0
,
1
12
)
+
ζ
″
(
0
,
11
12
)
}
+
4
π
ln
Γ
(
1
4
)
+
3
π
3
ln
Γ
(
1
3
)
−
{
2
+
3
2
π
+
3
2
ln
3
−
3
(
1
−
3
)
ln
2
+
2
3
ln
(
1
+
3
)
}
⋅
γ
−
2
3
(
3
ln
2
+
ln
3
+
ln
π
)
⋅
ln
(
1
+
3
)
−
7
−
6
3
2
ln
2
2
−
3
4
ln
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3
3
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1
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ln
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17
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2
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ln
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−
π
3
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2
+
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π
=
−
29
,
842878232041
…
,
{\displaystyle {\begin{aligned}\displaystyle \gamma _{1}{\biggl (}\!{\frac {1}{12}}\!{\biggr )}=&\displaystyle \,\,\,\gamma _{1}+{\sqrt {3}}\left\{\zeta ''\!\left(\!0,\,{\frac {1}{12}}\!\right)+\zeta ''\!\left(\!0,\,{\frac {11}{12}}\right)\!\right\}+4\pi \ln \Gamma {\biggl (}\!{\frac {1}{4}}\!{\biggr )}+3\pi {\sqrt {3}}\ln \Gamma {\biggl (}\!{\frac {1}{3}}\!{\biggr )}\\[5mm]&\displaystyle -\left\{\!{\frac {2+{\sqrt {3}}}{2}}\pi +{\frac {3}{2}}\ln 3-{\sqrt {3}}(1-{\sqrt {3}})\ln {2}+2{\sqrt {3}}\ln \!{\big (}1+{\sqrt {3}}{\big )}\!\right\}\!\cdot \gamma \\[5mm]&\displaystyle -2{\sqrt {3}}{\big (}3\ln 2+\ln 3+\ln \pi {\big )}\!\cdot \ln \!{\big (}1+{\sqrt {3}})-{\frac {7-6{\sqrt {3}}}{2}}\ln ^{2}\!2-{\frac {3}{4}}\ln ^{2}\!3\\[5mm]&\displaystyle +{\frac {3{\sqrt {3}}(1-{\sqrt {3}})}{2}}\ln 3\cdot \ln 2+{\sqrt {3}}\ln 2\cdot \ln \pi -{\frac {\pi {\big (}17+8{\sqrt {3}}{\big )}}{2{\sqrt {3}}}}\ln 2\\[5mm]&\displaystyle +{\frac {\pi {\big (}1-{\sqrt {3}}{\big )}{\sqrt {3}}}{4}}\ln 3-\pi {\sqrt {3}}(2+{\sqrt {3}})\ln \pi =-29,842878232041\ldots \!\,,\end{aligned}}}
(OEIS A255189 ),
kakor tudi nekatere druge vrednosti.
Druga posplošena Stieltjesova konstanta [ uredi | uredi kodo ]
Drugo posplošeno Stieltjesovo konstanto so manj raziskovali od prve. Blagouchine je pokazal, da se lahko podobno kot prva posplošena Stieltjesova konstanta druga posplošena Stieltjesova konstanta z racionalnim argumentom izračuna s pomočjo formule:
γ
2
(
r
m
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2
3
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m
−
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2
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{\displaystyle {\begin{array}{rl}\displaystyle \gamma _{2}{\biggl (}{\frac {r}{m}}{\biggr )}=\,\gamma _{2}+{\frac {2}{3}}\!\sum _{l=1}^{m-1}\cos {\frac {2\pi rl}{m}}\cdot \zeta '''\!\left(\!0,\,{\frac {l}{m}}\!\right)-2(\gamma +\ln 2\pi m)\!\sum _{l=1}^{m-1}\cos {\frac {2\pi rl}{m}}\cdot \zeta ''\!\left(\!0,\,{\frac {l}{m}}\!\right)\\[6mm]\displaystyle \quad +\pi \!\sum _{l=1}^{m-1}\sin {\frac {2\pi rl}{m}}\cdot \zeta ''\!\left(\!0,\,{\frac {l}{m}}\!\right)-2\pi (\gamma +\ln 2\pi m)\!\sum _{l=1}^{m-1}\sin {\frac {2\pi rl}{m}}\cdot \ln \Gamma {\biggl (}{\frac {l}{m}}{\biggr )}-2\gamma _{1}\ln {m}\\[6mm]\displaystyle \quad -\gamma ^{3}-\left[(\gamma +\ln 2\pi m)^{2}-{\frac {\pi ^{2}}{12}}\right]\!\cdot \!\digamma \!{\biggl (}{\frac {r}{m}}{\biggr )}+{\frac {\pi ^{3}}{12}}\cot {\frac {\pi r}{m}}-\gamma ^{2}\ln {\big (}4\pi ^{2}m^{3}{\big )}+{\frac {\pi ^{2}}{12}}(\gamma +\ln {m})\\[6mm]\displaystyle \quad -\gamma {\big (}\ln ^{2}\!{2\pi }+4\ln {m}\cdot \ln {2\pi }+2\ln ^{2}\!{m}{\big )}-\left\{\!\ln ^{2}\!{2\pi }+2\ln {2\pi }\cdot \ln {m}+{\frac {2}{3}}\ln ^{2}\!{m}\!\right\}\!\ln {m}\end{array}}\,,\qquad \quad (r=1,2,3,\ldots ,m-1)\!\,.}
Podobni rezultat je kasneje dobil Coffey z drugo metodo.[27]
↑ V primeru
n
=
0
{\displaystyle n=0\,}
prvi sumandd zahteva računanje 00 , kar je zaradi praznega produkta po dogovoru enako 1 .
↑ Glej seznam virov dan v [8] .
↑ Za podrobnosti in druge sumacijske formule glej [7] [24] .
↑ 1,0 1,1 Adell (2011) .
↑ Stieltjes (1905) .
↑ 3,0 3,1 Coppo (1999) .
↑ 4,0 4,1 Coffey (2009) .
↑ 5,0 5,1 Coffey (2010) .
↑ Choi (2013) .
↑ 7,0 7,1 7,2 7,3 7,4 7,5 7,6 Blagouchine (2015a) .
↑ 8,0 8,1 8,2 8,3 Blagouchine (2015b) .
↑ »Math StackExchange: A couple of definite integrals related to Stieltjes constants« (v angleščini).
↑ Hardy (2012) .
↑ Israilov (1981) .
↑ »Math StackExchange: A closed form for the series ...« (v angleščini).
↑ Berndt (1972) .
↑ Matsuoka (1985) .
↑ 15,0 15,1 15,2 Knessl; Coffey (2011) .
↑ Fekih-Ahmed (2014) .
↑ Keiper (1992) .
↑ Kreminski (2003) .
↑ Plouffe (1986) .
↑ Johansson (2013) .
↑ »Stieltjes Constants« . LMFDB (v angleščini). 5. avgust 2015. Pridobljeno 7. avgusta 2015 .
↑ 22,0 22,1 »Math StackExchange: Definite integral« (v angleščini).
↑ Connon (2009a) .
↑ 24,0 24,1 24,2 24,3 Blagouchine (2014) .
↑ Adamchik (1997) .
↑ »Math StackExchange: evaluation of a particular integral« (v angleščini).
↑ 27,0 27,1 Coffey (2014) .
↑ Connon (2009b) .
Adamchik, V. (1997), »A class of logarithmic integrals«, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation : 1–8
Adell, José Antonio (2011), »Asymptotic estimates for Stieltjes constants: a probabilistic approach« (PDF) , Proc. R. Soc. A , 467 (2128): 954–963, doi :10.1098/rspa.2010.0397 , arhivirano iz prvotnega spletišča (PDF) dne 4. februarja 2012, pridobljeno 8. avgusta 2015
Berndt, Bruce Carl (1972), »On the Hurwitz Zeta-function«, Rocky Mountain Journal of Mathematics , 2 (1): 151–157
Blagouchine, Iaroslav V. (2014), »Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results« , The Ramanujan Journal , 35 (1): 21–110 PDF
Blagouchine, Iaroslav V. (2015a), »A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations« , J. Number Theory (Elsevier) , 148, 151: 537–592, 276–277, arXiv :1401.3724
Blagouchine, Iaroslav V. (2015b), »Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only« , J. Number Theory (Elsevier) , 158 : 365–396, arXiv :1501.00740
Choi, Junesang (2013), »Certain integral representations of Stieltjes constants«, Journal of Inequalities and Applications , 532 : 1–10
Coffey, Mark W. (2009), Series representations for the Stieltjes constants , arXiv :0905.1111
Coffey, Mark W. (2010), »Addison-type series representation for the Stieltjes constants« , J. Number Theory , 130 : 2049–2064
Coffey, Mark W. (2014), Functional equations for the Stieltjes constants , arXiv :1402.3746
Connon, Donal F. (2009a), New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions , arXiv :0903.4539
Connon, Donal F. (2009b), The difference between two Stieltjes constants , arXiv :0906.0277
Coppo, Marc-Antoine (1999), »Nouvelles expressions des constantes de Stieltjes«, Expositiones Mathematicae , 17 : 349–358
Fekih-Ahmed, Lazhar (2014), A New Effective Asymptotic Formula for the Stieltjes Constants , arXiv :1407.5567
Hardy, Godfrey Harold (2012), »Note on Dr. Vacca's series for γ«, Q. J. Pure Appl. Math. , 43 : 215–216
Israilov, Maruf Israilovič (1981), »On the Laurent decomposition of Riemann's zeta function [v ruščini]« , Trudy Mat. Inst. Akad. Nauk. SSSR , 158 : 98–103, MR 0662837 , Zbl 0477.10031
Johansson, Fredrik (2013), Rigorous high-precision computation of the Hurwitz zeta function and its derivatives , arXiv :1309.2877
Keiper, Jerry B. (1992), »Power series expansions of Riemann ζ-function«, Math. Comp. , 58 (198): 765–773, Bibcode :1992MaCom..58..765K , doi :10.1090/S0025-5718-1992-1122072-5
Knessl, Charles; Coffey, Mark W. (2011), »An effective asymptotic formula for the Stieltjes constants«, Math. Comp. , 80 (273): 379–386
Kreminski, Rick (2003), »Newton-Cotes integration for approximating Stieltjes generalized Euler constants«, Math. Comp. , 72 (243): 1379–1397
Matsuoka, Yasushi (1985), »Generalized Euler Constants Associated with the Riemann Zeta Function«, Number Theory and Combinatorics , Singapur: World Scientific, str. 279–295, MR 0827790
Plouffe, Simon (1986), Stieltjes Constants, from 0 to 78, 256 digits each (v angleščini), pridobljeno 7. avgusta 2015
Stieltjes, Thomas Joannes (1905), Correspondance d'Hermite et de Stieltjes, volumes 1, 2 , Pariz: Gauthier-Villars