Le costanti di Stieltjes sono date dal limite
γ
n
=
lim
m
→
∞
{
∑
k
=
1
m
(
ln
k
)
n
k
−
(
ln
m
)
n
+
1
n
+
1
}
,
{\displaystyle \gamma _{n}=\lim _{m\rightarrow \infty }{\left\{\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}-{\frac {(\ln m)^{n+1}}{n+1}}\right\}},}
nel caso
n
=
0
{\displaystyle n=0}
0
0
{\displaystyle 0^{0}}
La formula di Cauchy fornisce una rappresentazione integrale
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)dx.}
Altre rappresentazioni in termini di serie e integrali compaiono nei lavori di Jensen , Franel, Hermite , Hardy , Ramanujan , Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine e altri autori.[ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8]
γ
n
=
1
2
δ
n
,
0
+
1
i
∫
0
∞
d
x
e
2
π
x
−
1
{
(
ln
(
1
−
i
x
)
)
n
1
−
i
x
−
(
ln
(
1
+
i
x
)
)
n
1
+
i
x
}
,
n
=
0
,
1
,
2
,
…
{\displaystyle \gamma _{n}={\frac {1}{2}}\delta _{n,0}+{\frac {1}{i}}\int _{0}^{\infty }{\frac {dx}{e^{2\pi x}-1}}\left\{{\frac {(\ln(1-ix))^{n}}{1-ix}}-{\frac {(\ln(1+ix))^{n}}{1+ix}}\right\}\,,\qquad \quad n=0,1,2,\ldots }
dove
δ
n
,
k
{\displaystyle \delta _{n,k}}
delta di Kronecker .[ 5] [ 7] [ 1] [ 5] [ 9]
γ
n
=
−
π
2
(
n
+
1
)
∫
−
∞
∞
(
ln
(
1
2
±
i
x
)
)
n
+
1
cosh
2
π
x
d
x
n
=
0
,
1
,
2
,
…
{\displaystyle \gamma _{n}=-{\frac {\pi }{2(n+1)}}\int _{-\infty }^{\infty }{\frac {\left(\ln \left({\frac {1}{2}}\pm ix\right)\right)^{n+1}}{\cosh ^{2}\pi x}}\,dx\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 {dx}{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\,dx.\end{array}}}
Per quanto riguarda le rappresentazioni in serie, nel 1912 Hardy [ 10]
γ
1
=
ln
2
2
∑
k
=
2
∞
(
−
1
)
k
k
⌊
log
2
k
⌋
⋅
(
2
log
2
k
−
⌊
log
2
2
k
⌋
)
.
{\displaystyle \gamma _{1}={\frac {\ln 2}{2}}\sum _{k=2}^{\infty }{\frac {(-1)^{k}}{k}}\lfloor \log _{2}{k}\rfloor \cdot \left(2\log _{2}{k}-\lfloor \log _{2}{2k}\rfloor \right).}
Israilov[ 11] numeri di Bernoulli
B
2
k
{\displaystyle B_{2k}}
γ
m
=
∑
k
=
1
n
(
ln
k
)
m
k
−
(
ln
n
)
m
+
1
m
+
1
−
(
ln
n
)
m
2
n
−
∑
k
=
1
N
−
1
B
2
k
(
2
k
)
!
[
(
ln
x
)
m
x
]
x
=
n
(
2
k
−
1
)
−
θ
⋅
B
2
N
(
2
N
)
!
[
(
ln
x
)
m
x
]
x
=
n
(
2
N
−
1
)
,
0
<
θ
<
1
{\displaystyle \gamma _{m}=\sum _{k=1}^{n}{\frac {(\ln k)^{m}}{k}}-{\frac {(\ln n)^{m+1}}{m+1}}-{\frac {(\ln n)^{m}}{2n}}-\sum _{k=1}^{N-1}{\frac {B_{2k}}{(2k)!}}\left[{\frac {(\ln x)^{m}}{x}}\right]_{x=n}^{(2k-1)}-\theta \cdot {\frac {B_{2N}}{(2N)!}}\left[{\frac {(\ln x)^{m}}{x}}\right]_{x=n}^{(2N-1)}\,,\qquad 0<\theta <1}
Connon,[ 12] [ 7] [ 1] coefficienti binomiali
γ
m
=
−
1
m
+
1
∑
n
=
0
∞
1
n
+
1
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
ln
(
k
+
1
)
)
m
+
1
γ
m
=
−
1
m
+
1
∑
n
=
0
∞
1
n
+
2
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
ln
(
k
+
1
)
)
m
+
1
k
+
1
γ
m
=
∑
n
=
0
∞
|
G
n
+
1
|
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
ln
(
k
+
1
)
)
m
k
+
1
{\displaystyle {\begin{array}{l}\displaystyle \gamma _{m}=-{\frac {1}{m+1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(\ln(k+1))^{m+1}\\[7mm]\displaystyle \gamma _{m}=-{\frac {1}{m+1}}\sum _{n=0}^{\infty }{\frac {1}{n+2}}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {(\ln(k+1))^{m+1}}{k+1}}\\[7mm]\displaystyle \gamma _{m}=\sum _{n=0}^{\infty }\left|G_{n+1}\right|\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {(\ln(k+1))^{m}}{k+1}}\end{array}}}
dove
G
n
{\displaystyle G_{n}}
coefficienti di Gregory , anche conosciuti come numeri logaritmici reciproci[ 13]
G
1
=
+
1
/
2
{\displaystyle G_{1}=+1/2}
G
2
=
−
1
/
12
{\displaystyle G_{2}=-1/12}
G
3
=
+
1
/
24
{\displaystyle G_{3}=+1/24}
G
4
=
−
19
/
720
{\displaystyle G_{4}=-19/720}
[ 14] numeri armonici conducono alle costanti di Stieltjes
∑
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[ 7]
[
⋅
⋅
]
{\displaystyle \left[{\cdot \atop \cdot }\right]}
γ
m
=
1
2
δ
m
,
0
+
(
−
1
)
m
m
!
π
∑
n
=
1
∞
1
n
⋅
n
!
∑
k
=
0
⌊
n
/
2
⌋
(
−
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 n/2\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,...,}
insieme a una serie semiconvergente con solo termini razionali
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
,
{\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,}
dove
M
=
0
,
1
,
2
,
…
{\displaystyle M=0,1,2,\ldots }
γ
1
=
−
1
2
∑
k
=
1
N
B
2
k
⋅
H
2
k
−
1
k
+
θ
⋅
B
2
N
+
2
⋅
H
2
N
+
1
2
N
+
2
,
0
<
θ
<
1
,
{\displaystyle \gamma _{1}=-{\frac {1}{2}}\sum _{k=1}^{N}{\frac {B_{2k}\cdot H_{2k-1}}{k}}+\theta \cdot {\frac {B_{2N+2}\cdot H_{2N+1}}{2N+2}},\qquad 0<\theta <1,}
dove
H
n
{\displaystyle H_{n}}
n
{\displaystyle n}
numero armonico .[ 7] [ 2] [ 3] [ 7]
Le costanti di Stieltjes in valore assoluto soddisfano il seguente maggiorante
|
γ
n
|
≤
{
2
(
n
−
1
)
!
π
n
,
n
=
1
,
3
,
5
,
…
4
(
n
−
1
)
!
π
n
,
n
=
2
,
4
,
6
,
…
{\displaystyle |\gamma _{n}|\leq {\begin{cases}\displaystyle {\frac {2(n-1)!}{\pi ^{n}}}\,,\qquad &n=1,3,5,\ldots \\[3mm]\displaystyle {\frac {4(n-1)!}{\pi ^{n}}}\,,\qquad &n=2,4,6,\ldots \end{cases}}}
scoperto da Berndt nel 1972.[ 15] [ 16]
|
γ
n
|
≤
n
!
2
n
+
1
,
n
=
1
,
2
,
3
,
…
{\displaystyle |\gamma _{n}|\leq {\frac {n!}{2^{n+1}}},\qquad n=1,2,3,\ldots }
e da Israilov[ 11]
|
γ
n
|
≤
n
!
C
(
k
)
(
2
k
)
n
,
n
=
1
,
2
,
3
,
…
{\displaystyle |\gamma _{n}|\leq {\frac {n!C(k)}{(2k)^{n}}},\qquad n=1,2,3,\ldots }
con
k
=
1
,
2
,
…
{\displaystyle k=1,2,\ldots }
C
(
1
)
=
1
/
2
{\displaystyle C(1)=1/2}
C
(
2
)
=
7
/
12
{\displaystyle C(2)=7/12}
[ 17]
|
γ
n
|
≤
{
2
(
2
n
)
!
n
n
+
1
(
2
π
)
n
,
n
=
1
,
3
,
5
,
…
4
(
2
n
)
!
n
n
+
1
(
2
π
)
n
,
n
=
2
,
4
,
6
,
…
{\displaystyle |\gamma _{n}|\leq {\begin{cases}\displaystyle {\frac {2(2n)!}{n^{n+1}(2\pi )^{n}}}\,,\qquad &n=1,3,5,\ldots \\[4mm]\displaystyle {\frac {4(2n)!}{n^{n+1}(2\pi )^{n}}}\,,\qquad &n=2,4,6,\ldots \end{cases}}}
e inoltre da Blagouchine[ 7]
−
|
B
m
+
1
|
m
+
1
<
γ
m
<
(
3
m
+
8
)
⋅
|
B
m
+
3
|
24
−
|
B
m
+
1
|
m
+
1
,
m
=
1
,
5
,
9
,
…
|
B
m
+
1
|
m
+
1
−
(
3
m
+
8
)
⋅
|
B
m
+
3
|
24
<
γ
m
<
|
B
m
+
1
|
m
+
1
,
m
=
3
,
7
,
11
,
…
−
|
B
m
+
2
|
2
<
γ
m
<
(
m
+
3
)
(
m
+
4
)
⋅
|
B
m
+
4
|
48
−
|
B
m
+
2
|
2
,
m
=
2
,
6
,
10
,
…
|
B
m
+
2
|
2
−
(
m
+
3
)
(
m
+
4
)
⋅
|
B
m
+
4
|
48
<
γ
m
<
|
B
m
+
2
|
2
,
m
=
4
,
8
,
12
,
…
{\displaystyle {\begin{array}{ll}\displaystyle -{\frac {{\big |}{B}_{m+1}{\big |}}{m+1}}<\gamma _{m}<{\frac {(3m+8)\cdot {\big |}{B}_{m+3}{\big |}}{24}}-{\frac {{\big |}{B}_{m+1}{\big |}}{m+1}},&m=1,5,9,\ldots \\[12pt]\displaystyle {\frac {{\big |}B_{m+1}{\big |}}{m+1}}-{\frac {(3m+8)\cdot {\big |}B_{m+3}{\big |}}{24}}<\gamma _{m}<{\frac {{\big |}{B}_{m+1}{\big |}}{m+1}},&m=3,7,11,\ldots \\[12pt]\displaystyle -{\frac {{\big |}{B}_{m+2}{\big |}}{2}}<\gamma _{m}<{\frac {(m+3)(m+4)\cdot {\big |}{B}_{m+4}{\big |}}{48}}-{\frac {{\big |}B_{m+2}{\big |}}{2}},\qquad &m=2,6,10,\ldots \\[12pt]\displaystyle {\frac {{\big |}{B}_{m+2}{\big |}}{2}}-{\frac {(m+3)(m+4)\cdot {\big |}{B}_{m+4}{\big |}}{48}}<\gamma _{m}<{\frac {{\big |}{B}_{m+2}{\big |}}{2}},&m=4,8,12,\ldots \\\end{array}}}
dove
B
n
{\displaystyle B_{n}}
numeri di Bernoulli . Infine si ha la seguente stima di Matsuoka[ 18] [ 19]
|
γ
n
|
<
10
−
4
e
n
ln
ln
n
,
n
=
5
,
6
,
7
,
…
{\displaystyle |\gamma _{n}|<10^{-4}e^{n\ln \ln n}\,,\qquad n=5,6,7,\ldots }
Per quanto riguarda le stime per mezzo di funzioni non elementari e loro soluzioni, Knessl, Coffey[ 20] [ 21]
n
{\displaystyle n}
[ 20]
v
{\displaystyle v}
2
π
exp
(
v
tan
v
)
=
n
cos
(
v
)
v
{\displaystyle 2\pi \exp(v\tan v)=n{\frac {\cos(v)}{v}}}
con
0
<
v
<
π
/
2
{\displaystyle 0<v<\pi /2}
u
=
v
tan
v
{\displaystyle u=v\tan v}
γ
n
∼
B
n
e
n
A
cos
(
a
n
+
b
)
{\displaystyle \gamma _{n}\sim {\frac {B}{\sqrt {n}}}e^{nA}\cos(an+b)}
dove
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
=
tan
−
1
(
v
u
)
+
v
u
2
+
v
2
{\displaystyle a=\tan ^{-1}\left({\frac {v}{u}}\right)+{\frac {v}{u^{2}+v^{2}}}}
b
=
tan
−
1
(
v
u
)
−
1
2
(
v
u
+
1
)
.
{\displaystyle b=\tan ^{-1}\left({\frac {v}{u}}\right)-{\frac {1}{2}}\left({\frac {v}{u+1}}\right).}
Fino a
n
=
100000
{\displaystyle n=100000}
γ
n
{\displaystyle \gamma _{n}}
n
=
137
{\displaystyle n=137}
[ 20]
I primi valori di
γ
n
{\displaystyle \gamma _{n}}
n
{\displaystyle n}
valore approssimato di
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
Per
n
{\displaystyle n}
Si possono trovare ulteriori informazioni sulla valutazione numerica delle costanti di Stieltjes nei lavori di Keiper,[ 22] [ 23] [ 24] [ 25]
n
=
100000
{\displaystyle n=100000}
[ 26]
Costanti di Stieltjes generalizzate
modifica
Più in generale, si possono definire le costanti di Stieltjes
γ
n
(
a
)
{\displaystyle \gamma _{n}(a)}
serie di Laurent della funzione zeta di Hurwitz :
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},}
dove
a
{\displaystyle a}
numero complesso con
Re
(
a
)
>
0
{\displaystyle \operatorname {Re} (a)>0}
γ
n
(
1
)
=
γ
n
{\displaystyle \gamma _{n}(1)=\gamma _{n}}
n
=
0
{\displaystyle n=0}
funzione digamma
γ
0
(
a
)
=
−
ψ
(
a
)
{\displaystyle \gamma _{0}(a)=-\psi (a)}
[ 27]
γ
n
(
a
)
=
lim
m
→
∞
{
∑
k
=
0
m
(
ln
(
k
+
a
)
)
n
k
+
a
−
(
ln
(
m
+
a
)
)
n
+
1
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(k+a))^{n}}{k+a}}-{\frac {(\ln(m+a))^{n+1}}{n+1}}\right\},\qquad {\begin{array}{l}n=0,1,2,\ldots \\[1mm]a\neq 0,-1,-2,\ldots \end{array}}}
dovuta a Berndt e Wilton. L'analoga per le costanti di Stieltjes generalizzate della formula di Jensen-Franel è la formula di Hermite [ 5]
γ
n
(
a
)
=
[
1
2
a
−
ln
a
n
+
1
]
(
ln
a
)
n
−
i
∫
0
∞
d
x
e
2
π
x
−
1
{
(
ln
(
a
−
i
x
)
)
n
a
−
i
x
−
(
ln
(
a
+
i
x
)
)
n
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 a)^{n}-i\int _{0}^{\infty }{\frac {dx}{e^{2\pi x}-1}}\left\{{\frac {(\ln(a-ix))^{n}}{a-ix}}-{\frac {(\ln(a+ix))^{n}}{a+ix}}\right\},\qquad {\begin{array}{l}n=0,1,2,\ldots \\[1mm]a\neq 0,-1,-2,\ldots \end{array}}}
Le costanti di Stieltjes generalizzate soddisfano la seguente relazione ricorsiva
γ
n
(
a
+
1
)
=
γ
n
(
a
)
−
(
ln
a
)
n
a
,
n
=
0
,
1
,
2
,
…
a
≠
0
,
−
1
,
−
2
,
…
{\displaystyle \gamma _{n}(a+1)=\gamma _{n}(a)-{\frac {(\ln a)^{n}}{a}}\,,\qquad {\begin{array}{l}n=0,1,2,\ldots \\[1mm]a\neq 0,-1,-2,\ldots \end{array}}}
come anche il teorema di moltiplicazione
∑
l
=
0
n
−
1
γ
p
(
a
+
l
n
)
=
(
−
1
)
p
n
[
ln
n
p
+
1
−
Ψ
(
a
n
)
]
(
ln
n
)
p
+
n
∑
r
=
0
p
−
1
(
−
1
)
r
(
p
r
)
γ
p
−
r
(
a
n
)
⋅
(
ln
n
)
r
,
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}}-\Psi (an)\right](\ln n)^{p}+n\sum _{r=0}^{p-1}(-1)^{r}{\binom {p}{r}}\gamma _{p-r}(an)\cdot (\ln n)^{r}\,,\qquad \qquad n=2,3,4,\ldots }
dove
(
p
r
)
{\displaystyle {\binom {p}{r}}}
coefficiente binomiale .[ 28] [ 29]
Prima costante di Stieltjes generalizzata
modifica
La prima costante di Stieltjes generalizzata possiede un bel numero di proprietà notevoli.
Identità di Malmsten (formula di riflessione per la prima costante generalizzata): la formula di riflessione per la prima costante di Stieltjes generalizzata è la seguente
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}}}
dove
m
{\displaystyle m}
n
{\displaystyle n}
m
<
n
{\displaystyle m<n}
[ 30] Carl Malmsten nel 1846.[ 5] [ 31]
Teorema degli argomenti razionali: la prima costante di Stieltjes generalizzata può essere calcolata nei numeri razionali in forma quasi-chiusa attraverso la seguente formula[ 5] [ 27]
γ
1
(
r
m
)
=
γ
1
+
γ
2
+
γ
ln
2
π
m
+
ln
2
π
⋅
ln
m
+
1
2
(
ln
m
)
2
+
(
γ
+
ln
2
π
m
)
⋅
Ψ
(
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 m)^{2}+(\gamma +\ln 2\pi m)\cdot \Psi \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.}
Una dimostrazione alternativa fu successivamente proposta da Coffey[ 32]
Somme finite: esistono molte sommatorie che riguardano la prima costante di Stieltjes generalizzata. Per esempio:
∑
r
=
0
m
−
1
γ
1
(
a
+
r
m
)
=
m
ln
m
⋅
Ψ
(
a
m
)
−
m
2
(
ln
m
)
2
+
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
m
)
2
∑
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
m
)
2
}
−
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 \Psi (am)-{\frac {m}{2}}(\ln m)^{2}+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 m)^{2}\\[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 m)^{2}\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}}}
Per ulteriori dettagli e sommatorie, vedere[ 5] [ 29]
Alcuni valori particolari: Alcuni particolari valori di
γ
1
(
a
)
{\displaystyle \gamma _{1}(a)}
a
{\displaystyle a}
funzione gamma , alla prima costante di Stieltjes e a qualche funzione elementare. Per esempio,
γ
1
(
1
2
)
=
−
2
γ
ln
2
−
(
ln
2
)
2
+
γ
1
=
−
1
,
353459680
…
{\displaystyle \gamma _{1}\left({\frac {1}{2}}\right)=-2\gamma \ln 2-(\ln 2)^{2}+\gamma _{1}=-1,353459680\ldots }
I valori della prima costante di Stieltjes generalizzata nei punti
1
/
4
{\displaystyle 1/4}
3
/
4
{\displaystyle 3/4}
1
/
3
{\displaystyle 1/3}
[ 33] [ 29]
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
,
518076350
…
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
,
3912989024
…
γ
1
(
1
3
)
=
−
3
γ
2
ln
3
−
3
4
(
ln
3
)
2
+
π
4
3
{
ln
3
−
8
ln
2
π
−
2
γ
+
12
ln
Γ
(
1
3
)
}
+
γ
1
=
−
3
,
259557515
…
{\displaystyle {\begin{array}{l}\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}=-5,518076350\ldots \\[6mm]\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}=-0,3912989024\ldots \\[6mm]\displaystyle \gamma _{1}\left({\frac {1}{3}}\right)=-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}(\ln 3)^{2}+{\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}=-3,259557515\ldots \end{array}}}
Nei punti
2
/
3
{\displaystyle 2/3}
1
/
6
{\displaystyle 1/6}
5
/
6
{\displaystyle 5/6}
γ
1
(
2
3
)
=
−
3
γ
2
ln
3
−
3
4
(
ln
3
)
2
−
π
4
3
{
ln
3
−
8
ln
2
π
−
2
γ
+
12
ln
Γ
(
1
3
)
}
+
γ
1
=
−
0
,
5989062842
…
γ
1
(
1
6
)
=
−
3
γ
2
ln
3
−
3
4
(
ln
3
)
2
−
(
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
,
74258252
…
γ
1
(
5
6
)
=
−
3
γ
2
ln
3
−
3
4
(
ln
3
)
2
−
(
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
,
2461690038
…
{\displaystyle {\begin{array}{l}\displaystyle \gamma _{1}\left({\frac {2}{3}}\right)=-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}(\ln 3)^{2}-{\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}=-0,5989062842\ldots \\[6mm]\displaystyle \gamma _{1}\left({\frac {1}{6}}\right)=-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}(\ln 3)^{2}-(\ln 2)^{2}-(3\ln 3+2\gamma )\ln 2+{\frac {3\pi {\sqrt {3}}}{2}}\ln \Gamma \left({\frac {1}{6}}\right)\\[5mm]\displaystyle \qquad \qquad \quad -{\frac {\pi }{2{\sqrt {3}}}}\left\{3\ln 3+11\ln 2+{\frac {15}{2}}\ln \pi +3\gamma \right\}+\gamma _{1}=-10,74258252\ldots \\[6mm]\displaystyle \gamma _{1}\left({\frac {5}{6}}\right)=-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}(\ln 3)^{2}-(\ln 2)^{2}-(3\ln 3+2\gamma )\ln 2-{\frac {3\pi {\sqrt {3}}}{2}}\ln \Gamma \left({\frac {1}{6}}\right)\\[6mm]\displaystyle \qquad \qquad \quad +{\frac {\pi }{2{\sqrt {3}}}}\left\{3\ln 3+11\ln 2+{\frac {15}{2}}\ln \pi +3\gamma \right\}+\gamma _{1}=-0,2461690038\ldots \end{array}}}
Questi valori sono stati calcolati da Blagouchine.[ 29]
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
5
)
2
+
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
,
030205511
…
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
,
64171976
…
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
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84287823
…
{\displaystyle {\begin{array}{ll}\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 5)^{2}\\[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 =-8,030205511\ldots \\[6mm]\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 =-16,64171976\ldots \\[6mm]\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 3)^{2}\\[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 =-29,84287823\ldots \end{array}}}
Seconda costante di Stieltjes generalizzata
modifica
La seconda costante di Stieltjes generalizzata è molto meno studiata della prima. Similmente alla prima, si possono calcolare i valori di
γ
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{\displaystyle \gamma _{2}(a)}
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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 \Psi {\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\pi )^{2}+4\ln m\cdot \ln 2\pi +2(\ln m)^{2}{\big )}-\left\{(\ln 2\pi )^{2}+2\ln 2\pi \cdot \ln m+{\frac {2}{3}}(\ln m)^{2}\right\}\ln m.\end{array}}}
Un risultato equivalente fu ottenuto successivamente da Coffey per mezzo di un altro metodo.[ 32]
^ a b c Marc-Antoine Coppo. Nouvelles expressions des constantes de Stieltjes . Expositiones Mathematicae, vol. 17, pp. 349-358, 1999.
^ a b Mark W. Coffey. Series representations for the Stieltjes constants , arXiv:0905.1111
^ a b (EN Addison-type series representation for the Stieltjes constants Journal of Number Theory , vol. 130, n. 9, 1º settembre 2010, pp. 2049–2064, DOI :10.1016/j.jnt.2010.01.003 . URL consultato il 13 settembre 2022 . ^ Junesang Choi. Certain integral representations of Stieltjes constants , Journal of Inequalities and Applications, 2013:532, pp. 1-10
^ a b c d e f g h
(EN A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations Journal of Number Theory , vol. 148, 1º marzo 2015, pp. 537–592, DOI :10.1016/j.jnt.2014.08.009 . URL consultato il 13 settembre 2022 .
^ Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations Journal of Number Theory , vol. 151, 2015-06, pp. 276–277, DOI :10.1016/J.JNT.2015.01.001 . URL consultato il 13 settembre 2022 . ^ a b c d e f g (EN Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only Journal of Number Theory , vol. 158, 1º gennaio 2016, pp. 365–396, DOI :10.1016/j.jnt.2015.06.012 . URL consultato il 13 settembre 2022 . ^ Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in $\pi^{-2}$ and into the formal enveloping series with rational coefficients only Journal of Number Theory , vol. 173, 2017-04, pp. 631–632, DOI :10.1016/J.JNT.2016.11.002 . URL consultato il 13 settembre 2022 . ^ (EN calculus - A couple of definite integrals related to Stieltjes constants Mathematics Stack Exchange . URL consultato il 13 settembre 2022 . ^ G. H. Hardy. Note on Dr. Vacca's series for γ , Q. J. Pure Appl. Math. 43, pp. 215–216, 2012.
^ a b M. I. Israilov. On the Laurent decomposition of Riemann's zeta function [in Russian] . Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 158, pp. 98-103, 1981.
^ Donal F. Connon, Some applications of the Stieltjes constants arXiv:0901.2083 [math] , 14 gennaio 2009. URL consultato il 13 settembre 2022 . ^ (EN Logarithmic Number mathworld.wolfram.com . URL consultato il 13 settembre 2022 . ^ (EN calculus - A closed form of the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$ Mathematics Stack Exchange . URL consultato il 13 settembre 2022 . ^ Bruce C. Berndt. On the Hurwitz Zeta-function . Rocky Mountain Journal of Mathematics, vol. 2, no. 1, pp. 151-157, 1972.
^ A. F. Lavrik. On the main term of the divisor's problem and the power series of the Riemann's zeta function in a neighbourhood of its pole (in Russian).Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 142, pp. 165-173, 1976.
^ Z. Nan-You and K. S. Williams. Some results on the generalized Stieltjes constants . Analysis, vol. 14, pp. 147-162, 1994.
^ Y. Matsuoka. Generalized Euler constants associated with the Riemann zeta function . Number Theory and Combinatorics: Japan 1984, World Scientific, Singapore, pp. 279-295, 1985
^ Y. Matsuoka. On the power series coefficients of the Riemann zeta function . Tokyo Journal of Mathematics, vol. 12, no. 1, pp. 49-58, 1989.
^ a b c Charles Knessl e Mark W. Coffey. An effective asymptotic formula for the Stieltjes constants . Math. Comp., vol. 80, no. 273, pp. 379-386, 2011.
^ Lazhar Fekih-Ahmed, A New Effective Asymptotic Formula for the Stieltjes Constants arXiv:1407.5567 [math] , 28 dicembre 2014. URL consultato il 13 settembre 2022 . ^ J.B. Keiper. Power series expansions of Riemann ζ-function . Math. Comp., vol. 58, no. 198, pp. 765-773, 1992.
^ Rick Kreminski. Newton-Cotes integration for approximating Stieltjes generalized Euler constants . Math. Comp., vol. 72, no. 243, pp. 1379-1397, 2003.
^ Simon Plouffe. Stieltjes Constants, from 0 to 78, 256 digits each ^ Fredrik Johansson. Rigorous high-precision computation of the Hurwitz zeta function and its derivatives , arXiv:1309.2877
^ LMFDB - Stieltjes Constants beta.lmfdb.org . URL consultato il 13 settembre 2022 . ^ a b (EN real analysis - Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$...Definite Integral Mathematics Stack Exchange . URL consultato il 13 settembre 2022 . ^ Donal F. Connon, New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions arXiv:0903.4539 [math] , 26 marzo 2009. URL consultato il 13 settembre 2022 . ^ a b c d (EN Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results The Ramanujan Journal , vol. 35, n. 1, 1º ottobre 2014, pp. 21–110, DOI :10.1007/s11139-013-9528-5 . URL consultato il 13 settembre 2022 . ^ V. Adamchik. A class of logarithmic integrals. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.
^ (EN calculus - Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$ Mathematics Stack Exchange . URL consultato il 13 settembre 2022 . ^ a b Mark W. Coffey, Functional equations for the Stieltjes constants arXiv:1402.3746 [math] , 16 maggio 2014. URL consultato il 13 settembre 2022 . ^ Donal F. Connon, The difference between two Stieltjes constants arXiv:0906.0277 [math] , 1º giugno 2009. URL consultato il 13 settembre 2022 . 
![{\displaystyle {\begin{array}{l}\displaystyle \gamma _{1}=-\left[\gamma -{\frac {\ln 2}{2}}\right]\ln 2+i\int _{0}^{\infty }{\frac {dx}{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\,dx.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c65016394c98204477eca6613715e19e7fc26360)
![{\displaystyle \gamma _{m}=\sum _{k=1}^{n}{\frac {(\ln k)^{m}}{k}}-{\frac {(\ln n)^{m+1}}{m+1}}-{\frac {(\ln n)^{m}}{2n}}-\sum _{k=1}^{N-1}{\frac {B_{2k}}{(2k)!}}\left[{\frac {(\ln x)^{m}}{x}}\right]_{x=n}^{(2k-1)}-\theta \cdot {\frac {B_{2N}}{(2N)!}}\left[{\frac {(\ln x)^{m}}{x}}\right]_{x=n}^{(2N-1)}\,,\qquad 0<\theta <1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82438cf84f77f8c4bf8899a6bb79b2478a91b290)
![{\displaystyle {\begin{array}{l}\displaystyle \gamma _{m}=-{\frac {1}{m+1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(\ln(k+1))^{m+1}\\[7mm]\displaystyle \gamma _{m}=-{\frac {1}{m+1}}\sum _{n=0}^{\infty }{\frac {1}{n+2}}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {(\ln(k+1))^{m+1}}{k+1}}\\[7mm]\displaystyle \gamma _{m}=\sum _{n=0}^{\infty }\left|G_{n+1}\right|\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {(\ln(k+1))^{m}}{k+1}}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c303f775ca32e85fa11b02c50fa31ee2edde8c94)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/281307bdf57df4c095995b391bf6d4715620c801)
![{\displaystyle \left[{\cdot \atop \cdot }\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e61b8728ed1e1a9034522fd06494ee3430e8fcd)
![{\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 n/2\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,...,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa52a594019bc77729f2efd85235db5f550d4f74)
![{\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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a3b8fd719684a18b9ae612fae4a3c16d7d7b34)
![{\displaystyle |\gamma _{n}|\leq {\begin{cases}\displaystyle {\frac {2(n-1)!}{\pi ^{n}}}\,,\qquad &n=1,3,5,\ldots \\[3mm]\displaystyle {\frac {4(n-1)!}{\pi ^{n}}}\,,\qquad &n=2,4,6,\ldots \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e537a5f5fd6c447939b3813b6ca963ec8dc385f2)
![{\displaystyle |\gamma _{n}|\leq {\begin{cases}\displaystyle {\frac {2(2n)!}{n^{n+1}(2\pi )^{n}}}\,,\qquad &n=1,3,5,\ldots \\[4mm]\displaystyle {\frac {4(2n)!}{n^{n+1}(2\pi )^{n}}}\,,\qquad &n=2,4,6,\ldots \end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3f0994e61e26db224a0c03cc9a5f4df8adb59d)
![{\displaystyle {\begin{array}{ll}\displaystyle -{\frac {{\big |}{B}_{m+1}{\big |}}{m+1}}<\gamma _{m}<{\frac {(3m+8)\cdot {\big |}{B}_{m+3}{\big |}}{24}}-{\frac {{\big |}{B}_{m+1}{\big |}}{m+1}},&m=1,5,9,\ldots \\[12pt]\displaystyle {\frac {{\big |}B_{m+1}{\big |}}{m+1}}-{\frac {(3m+8)\cdot {\big |}B_{m+3}{\big |}}{24}}<\gamma _{m}<{\frac {{\big |}{B}_{m+1}{\big |}}{m+1}},&m=3,7,11,\ldots \\[12pt]\displaystyle -{\frac {{\big |}{B}_{m+2}{\big |}}{2}}<\gamma _{m}<{\frac {(m+3)(m+4)\cdot {\big |}{B}_{m+4}{\big |}}{48}}-{\frac {{\big |}B_{m+2}{\big |}}{2}},\qquad &m=2,6,10,\ldots \\[12pt]\displaystyle {\frac {{\big |}{B}_{m+2}{\big |}}{2}}-{\frac {(m+3)(m+4)\cdot {\big |}{B}_{m+4}{\big |}}{48}}<\gamma _{m}<{\frac {{\big |}{B}_{m+2}{\big |}}{2}},&m=4,8,12,\ldots \\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11c9df6acbb388f10001103673363d0062133e6)
![{\displaystyle B={\frac {2{\sqrt {2\pi }}{\sqrt {u^{2}+v^{2}}}}{[(u+1)^{2}+v^{2}]^{1/4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/886d29018cfe3f8d545ca4beb14b1a38e5ee81a3)
![{\displaystyle \gamma _{n}(a)=\lim _{m\to \infty }\left\{\sum _{k=0}^{m}{\frac {(\ln(k+a))^{n}}{k+a}}-{\frac {(\ln(m+a))^{n+1}}{n+1}}\right\},\qquad {\begin{array}{l}n=0,1,2,\ldots \\[1mm]a\neq 0,-1,-2,\ldots \end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be51be89522ed2a53130f5b7a2faeb7ea0a137da)
^{n}-i\int _{0}^{\infty }{\frac {dx}{e^{2\pi x}-1}}\left\{{\frac {(\ln(a-ix))^{n}}{a-ix}}-{\frac {(\ln(a+ix))^{n}}{a+ix}}\right\},\qquad {\begin{array}{l}n=0,1,2,\ldots \\[1mm]a\neq 0,-1,-2,\ldots \end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/547003d58a8e3c1d3e4694bfb8e8c89b915bd1b0)
![{\displaystyle \gamma _{n}(a+1)=\gamma _{n}(a)-{\frac {(\ln a)^{n}}{a}}\,,\qquad {\begin{array}{l}n=0,1,2,\ldots \\[1mm]a\neq 0,-1,-2,\ldots \end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd753edda47e1b284d2f867e317fbff08f52793)
^{p}+n\sum _{r=0}^{p-1}(-1)^{r}{\binom {p}{r}}\gamma _{p-r}(an)\cdot (\ln n)^{r}\,,\qquad \qquad n=2,3,4,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d07a1295754761fd7e73c7e516d6bc5e80a2e05)
![{\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 m)^{2}+(\gamma +\ln 2\pi m)\cdot \Psi \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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2411b6ca2ff862c9967bf3deb3ad1577dbd6d9a7)
![{\displaystyle {\begin{array}{ll}\displaystyle \sum _{r=0}^{m-1}\gamma _{1}\left(a+{\frac {r}{m}}\right)=m\ln {m}\cdot \Psi (am)-{\frac {m}{2}}(\ln m)^{2}+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 m)^{2}\\[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 m)^{2}\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87199fb146cd466e8c061249c139ebea305df3ba)
![{\displaystyle {\begin{array}{l}\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}=-5,518076350\ldots \\[6mm]\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}=-0,3912989024\ldots \\[6mm]\displaystyle \gamma _{1}\left({\frac {1}{3}}\right)=-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}(\ln 3)^{2}+{\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}=-3,259557515\ldots \end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2770a30b93b451a8d810fbd33653e785fd7e1c79)
![{\displaystyle {\begin{array}{l}\displaystyle \gamma _{1}\left({\frac {2}{3}}\right)=-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}(\ln 3)^{2}-{\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}=-0,5989062842\ldots \\[6mm]\displaystyle \gamma _{1}\left({\frac {1}{6}}\right)=-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}(\ln 3)^{2}-(\ln 2)^{2}-(3\ln 3+2\gamma )\ln 2+{\frac {3\pi {\sqrt {3}}}{2}}\ln \Gamma \left({\frac {1}{6}}\right)\\[5mm]\displaystyle \qquad \qquad \quad -{\frac {\pi }{2{\sqrt {3}}}}\left\{3\ln 3+11\ln 2+{\frac {15}{2}}\ln \pi +3\gamma \right\}+\gamma _{1}=-10,74258252\ldots \\[6mm]\displaystyle \gamma _{1}\left({\frac {5}{6}}\right)=-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}(\ln 3)^{2}-(\ln 2)^{2}-(3\ln 3+2\gamma )\ln 2-{\frac {3\pi {\sqrt {3}}}{2}}\ln \Gamma \left({\frac {1}{6}}\right)\\[6mm]\displaystyle \qquad \qquad \quad +{\frac {\pi }{2{\sqrt {3}}}}\left\{3\ln 3+11\ln 2+{\frac {15}{2}}\ln \pi +3\gamma \right\}+\gamma _{1}=-0,2461690038\ldots \end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c82a4cda0beaedf2a53d6865d51c7ee604b19818)
![{\displaystyle {\begin{array}{ll}\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 5)^{2}\\[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 =-8,030205511\ldots \\[6mm]\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 =-16,64171976\ldots \\[6mm]\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 3)^{2}\\[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 =-29,84287823\ldots \end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ceaa2dada2937be1b54704e831cc905eb7c0cf)
![{\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 \Psi {\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\pi )^{2}+4\ln m\cdot \ln 2\pi +2(\ln m)^{2}{\big )}-\left\{(\ln 2\pi )^{2}+2\ln 2\pi \cdot \ln m+{\frac {2}{3}}(\ln m)^{2}\right\}\ln m.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11b7fd121062b8a60c4fb76bcca3ec0552b2f380)
