Stieltjesove konstante

Stieltjesove konstante (ali posplošene Eulerjeve konstante^[1]) so v matematiki števila ${\displaystyle \gamma _{n}\,}$, ki se pojavljajo v Laurentovi vrsti za Riemannovo funkcijo ζ:

${\displaystyle \zeta (s)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}\;(s-1)^{n}\!\,.}$

Ničta Stieltjesova konstanta ${\displaystyle \gamma _{0}\equiv \gamma ={0},577215664901\ldots \,}$ 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]

${\displaystyle \gamma _{n}=\lim _{m\rightarrow \infty }{\left\{\sum _{k=1}^{m}{\frac {\ln ^{n}k}{k}}-{\frac {\ln ^{n+1}\!m}{n+1}}\right\}}\!\,.}$^[a]

Cauchyjeva formula za odvod vodi do integralskega izraza:

${\displaystyle \gamma _{n}={\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)\mathrm {d} x\!\,.}$

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:

${\displaystyle \gamma _{n}\,=\,{\frac {1}{2}}\delta _{n,0}+\,{\frac {1}{i}}\!\int _{0}^{\infty }\!{\frac {\mathrm {d} x}{e^{2\pi x}-1}}\left\{{\frac {\ln ^{n}(1-ix)}{1-ix}}-{\frac {\ln ^{n}(1+ix)}{1+ix}}\right\}\,,\qquad (n=0,1,2,\ldots )\!\,,}$

kjer je ${\displaystyle \delta _{n,k}\,}$ Kroneckerjeva delta.^[7][8] Med drugimi formulami so (glej: ^[3][7][9]):

${\displaystyle \gamma _{n}\,=\,-{\frac {\pi }{2(n+1)}}\!\int _{-\infty }^{+\infty }{\frac {\ln ^{n+1}\!{\big (}{\frac {1}{2}}\pm ix{\big )}}{\cosh ^{2}\!\pi x}}\,\mathrm {d} x\,,\qquad \qquad \qquad \qquad \qquad (n=0,1,2,\ldots )\!\,,}$

${\displaystyle {\begin{array}{l}\displaystyle \gamma _{1}=-\left[\gamma -{\frac {\ln 2}{2}}\right]\ln 2+\,i\!\int _{0}^{\infty }\!{\frac {\mathrm {d} x}{e^{\pi x}+1}}\left\{{\frac {\ln(1-ix)}{1-ix}}-{\frac {\ln(1+ix)}{1+ix}}\right\}\,\\[6mm]\displaystyle \gamma _{1}=-\gamma ^{2}-\int _{0}^{\infty }\left[{\frac {1}{1-e^{-x}}}-{\frac {1}{x}}\right]e^{-x}\ln x\,\mathrm {d} x\!\,.\end{array}}}$

Znano vrsto, ki vsebuje celi del logaritma, je podal Hardy leta 1912:^[10]

${\displaystyle \gamma _{1}\,=\,{\frac {\ln 2}{2}}\sum _{k=2}^{\infty }{\frac {(-1)^{k}}{k}}\,\lfloor \operatorname {lb} k\rfloor \cdot {\big (}2\operatorname {lb} k-\lfloor \operatorname {lb} (2k)\rfloor {\big )}\!\,.}$

Tu je ${\displaystyle \operatorname {lb} \,}$ dvojiški logaritem.

Israilov je podal delno konvergentno vrsto z Bernoullijevimi števili ${\displaystyle B_{2k}\,}$:^[11]

${\displaystyle \gamma _{m}\,=\,\sum _{k=1}^{n}{\frac {\,\ln ^{m}\!k\,}{k}}-{\frac {\,\ln ^{m+1}\!n\,}{m+1}}-{\frac {\,\ln ^{m}\!n\,}{2n}}-\sum _{k=1}^{N-1}{\frac {\,B_{2k}\,}{(2k)!}}\left[{\frac {\ln ^{m}\!x}{x}}\right]_{x=n}^{(2k-1)}-\theta \cdot {\frac {\,B_{2N}\,}{(2N)!}}\left[{\frac {\ln ^{m}\!x}{x}}\right]_{x=n}^{(2N-1)}\,,\qquad (0<\theta <1)\!\,.}$

Oloa in Tauraso sta pokazala, da vrsta s harmoničnimi števili ${\displaystyle H_{n}\,}$ lahko vodi do Stieltjesovih konstant:^[12]

${\displaystyle {\begin{array}{l}\displaystyle \sum _{n=1}^{\infty }{\frac {\,H_{n}-(\gamma +\ln n)\,}{n}}\,=\,\,-\gamma _{1}-{\frac {1}{2}}\gamma ^{2}+{\frac {1}{12}}\pi ^{2}\\[6mm]\displaystyle \sum _{n=1}^{\infty }{\frac {\,H_{n}^{2}-(\gamma +\ln n)^{2}\,}{n}}\,=\,\,-\gamma _{2}-2\gamma \gamma _{1}-{\frac {2}{3}}\gamma ^{3}+{\frac {5}{3}}\zeta (3)\!\,.\end{array}}}$

Blagouchine je našel počasi konvergentno vrsto, ki vsebuje nepredznačena Stirlingova števila prve vrste ${\displaystyle \left[{\cdot \atop \cdot }\right]\,}$:^[8]

${\displaystyle \gamma _{m}\,=\,{\frac {1}{2}}\delta _{m,0}+{\frac {\,(-1)^{m}m!\,}{\pi }}\sum _{n=1}^{\infty }{\frac {1}{\,n\cdot n!\,}}\sum _{k=0}^{\lfloor \!{\frac {1}{2}}n\!\rfloor }{\frac {\,(-1)^{k}\cdot \left[{2k+2 \atop m+1}\right]\cdot \left[{n \atop 2k+1}\right]\,}{\,(2\pi )^{2k+1}\,}}\,,\qquad (m=0,1,2,\ldots )\!\,,}$

kot tudi delno konvergentno vrsto s samimi racionalnimi členi:

${\displaystyle \gamma _{m}\,=\,{\frac {1}{2}}\delta _{m,0}+(-1)^{m}m!\cdot \!\sum _{k=1}^{N}{\frac {\,\left[{2k \atop m+1}\right]\cdot B_{2k}\,}{(2k)!}}\,+\,\theta \cdot {\frac {\,(-1)^{m}m!\!\cdot \left[{2N+2 \atop m+1}\right]\cdot B_{2N+2}\,}{(2N+2)!}}\,,\qquad (0<\theta <1,m=0,1,2,\ldots )\!\,.}$

Več drugih vrst je danih v Coffeyjevemu delu.^[4][5]

Za Stieltjesove konstante velja meja:

${\displaystyle {\big |}\gamma _{n}{\big |}\,\leq \,{\begin{cases}\displaystyle {\frac {2\,(n-1)!}{\pi ^{n}}}\,;&n=1,3,5,\ldots \\[3mm]\displaystyle {\frac {4\,(n-1)!}{\pi ^{n}}}\,;&n=2,4,6,\ldots \!\,,\end{cases}}}$

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]

${\displaystyle |\gamma _{n}|<10^{-4}e^{n\ln \ln n}\,,\qquad (n\geq 5)\!\,.}$

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 ${\displaystyle n\,}$.^[15] Če je ${\displaystyle v\,}$ enolična rešitev enačbe:

${\displaystyle 2\pi \exp(v\operatorname {tg} \,v)=n{\frac {\cos v}{v}}\!\,,}$

z ${\displaystyle 0<v<\pi /2\,}$, in, če je ${\displaystyle u=v\operatorname {tg} \,v\,}$, potem velja:

${\displaystyle \gamma _{n}\sim {\frac {B}{\sqrt {n}}}e^{nA}\cos(an+b)\!\,,}$

kjer je:

${\displaystyle A={\frac {1}{2}}\ln(u^{2}+v^{2})-{\frac {u}{u^{2}+v^{2}}}\!\,,}$

${\displaystyle B={\frac {2{\sqrt {2\pi }}{\sqrt {u^{2}+v^{2}}}}{[(u+1)^{2}+v^{2}]^{1/4}}}\!\,,}$

${\displaystyle a=\operatorname {tg} ^{-1}\left({\frac {v}{u}}\right)+{\frac {v}{u^{2}+v^{2}}}\!\,,}$

${\displaystyle b=\operatorname {tg} ^{-1}\left({\frac {v}{u}}\right)-{\frac {1}{2}}\left({\frac {v}{u+1}}\right)\!\,.}$

Vse do ${\displaystyle n=100000\,}$ Knessl-Coffeyjev približek trenutno predvideva predznak ${\displaystyle \gamma _{n}\,}$ z eno izjemo za ${\displaystyle n=137\,}$.^[15]

Prve desetiške vrednosti Stieltjesovih konstant podaja razpredelnica:

${\displaystyle n\,}$	desetiške vrednosti ${\displaystyle \gamma _{n}\,}$	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 ${\displaystyle n\,}$ 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 ${\displaystyle n=100000\,}$, vsaka točna na več kot 10000 števk. Številske vrednosti se lahko dobijo v podatkovni bazi LMFDB.^[21]

Bolj splošno se lahko definirajo Stieltjesove konstante ${\displaystyle \gamma _{n}(a)\,}$, ki se pojavljajo v Laurentovi vrsti za Hurwitzevo funkcijo ζ:

${\displaystyle \zeta (s,a)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}(a)\;(s-1)^{n}\!\,.}$

Tu je ${\displaystyle a\,}$ kompleksno število z ${\displaystyle \Re (a)>0\,}$. Ker je Hurwitzeva funkcija ζ posplošitev Riemannove funkcije ζ, velja ${\displaystyle \gamma _{n}(1)=\gamma _{n}\,}$. Ničta konstanta je preprosto funkcija digama ${\displaystyle \gamma _{0}(a)=-\digamma (a)\,}$.^[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:

${\displaystyle \gamma _{n}(a)\,=\,\lim _{m\to \infty }\left\{\sum _{k=0}^{m}{\frac {\ln ^{n}(k+a)}{k+a}}-{\frac {\ln ^{n+1}(m+a)}{n+1}}\right\}\,,\qquad \;{\begin{array}{l}n=0,1,2,\ldots \,\\[1mm]a\neq 0,-1,-2,\ldots \!\,,\end{array}}}$

ki sta jo podala Berndt in Wilton. Analogon Jensen-Franelove formule za posplošeno Stieltjesovo konstanto je Hermitova formula:^[7]

${\displaystyle \gamma _{n}(a)\,=\,\left[{\frac {1}{2a}}-{\frac {\ln {a}}{n+1}}\right]\ln ^{n}\!{a}-i\!\int _{0}^{\infty }\!{\frac {\mathrm {d} x}{e^{2\pi x}-1}}\left\{{\frac {\ln ^{n}(a-ix)}{a-ix}}-{\frac {\ln ^{n}(a+ix)}{a+ix}}\right\}\,,\qquad \;{\begin{array}{l}n=0,1,2,\ldots \,\\[1mm]a\neq 0,-1,-2,\ldots \end{array}}}$

Za posplošene Stieltjesove konstante velja naslednja rekurenčna zveza:

${\displaystyle \gamma _{n}(a+1)\,=\,\gamma _{n}(a)-{\frac {\,\ln ^{n}\!a\,}{a}}\,,\qquad \;{\begin{array}{l}n=0,1,2,\ldots \,\\[1mm]a\neq 0,-1,-2,\ldots \!\,,\end{array}}}$

kakor tudi multiplikacijski izrek:

${\displaystyle \sum _{l=0}^{n-1}\gamma _{p}\!\left(\!a+{\frac {l}{\,n\,}}\right)=\,(-1)^{p}n\!\left[{\frac {\ln n}{\,p+1\,}}-\digamma (an)\right]\!\ln ^{p}\!n\,+\,n\sum _{r=0}^{p-1}(-1)^{r}{\binom {p}{r}}\gamma _{p-r}(an)\cdot \ln ^{r}\!{n}\,,\qquad (n=2,3,4,\ldots )\!\,,}$

kjer ${\displaystyle {\binom {p}{r}}}$ označuje binomski koeficient.^{[23][24]:101–102}

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:

${\displaystyle \gamma _{1}{\biggl (}{\frac {m}{n}}{\biggr )}-\gamma _{1}{\biggl (}1-{\frac {m}{n}}{\biggr )}=2\pi \sum _{l=1}^{n-1}\sin {\frac {2\pi ml}{n}}\cdot \ln \Gamma {\biggl (}{\frac {l}{n}}{\biggr )}-\pi (\gamma +\ln 2\pi n)\cot {\frac {m\pi }{n}}\!\,,}$

kjer sta ${\displaystyle m\,}$ in ${\displaystyle n\,}$ takšni pozitivni celi števii, da velja ${\displaystyle m<n\,}$, ${\displaystyle \Gamma \,}$ 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]

${\displaystyle {\begin{array}{ll}\displaystyle \gamma _{1}{\biggl (}{\frac {r}{m}}{\biggr )}=&\displaystyle \gamma _{1}+\gamma ^{2}+\gamma \ln 2\pi m+\ln 2\pi \cdot \ln {m}+{\frac {1}{2}}\ln ^{2}\!{m}+(\gamma +\ln 2\pi m)\cdot \digamma \!\left(\!{\frac {r}{m}}\!\right)\\[5mm]\displaystyle &\displaystyle \qquad +\pi \sum _{l=1}^{m-1}\sin {\frac {2\pi rl}{m}}\cdot \ln \Gamma {\biggl (}{\frac {l}{m}}{\biggr )}+\sum _{l=1}^{m-1}\cos {\frac {2\pi rl}{m}}\cdot \zeta ''\!\left(\!0,\,{\frac {l}{m}}\!\right)\end{array}}\,,\qquad \quad (r=1,2,3,\ldots ,m-1)\!\,.}$

Alternativni dokaz je kasneje predložil Coffey.^[27]

• Končne vsote: za prve posplošene Stieltjesove konstante obstaje veliko sumacijskih formul. Na primer:^[c]

${\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 ${\displaystyle \gamma _{1}\,}$ in elementarne funkcije. Na primer:

${\displaystyle \gamma _{1}\!\left(\!{\frac {1}{\,2\,}}\!\right)=-2\gamma \ln 2-\ln ^{2}\!2+\gamma _{1}\,=\,-1,353459680804\ldots \!\,,}$ (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]

${\displaystyle \displaystyle \gamma _{1}\!\left(\!{\frac {1}{\,4\,}}\!\right)=\,2\pi \ln \Gamma \!\left(\!{\frac {1}{\,4\,}}\!\right)-{\frac {3\pi }{2}}\ln \pi -{\frac {7}{2}}\ln ^{2}\!2-(3\gamma +2\pi )\ln 2-{\frac {\gamma \pi }{2}}+\gamma _{1}\,=\,-5,518076350199\ldots \!\,,}$ (OEIS A254347),

${\displaystyle \displaystyle \gamma _{1}\!\left(\!{\frac {3}{\,4\,}}\!\right)=\,-2\pi \ln \Gamma \!\left(\!{\frac {1}{\,4\,}}\!\right)+{\frac {3\pi }{2}}\ln \pi -{\frac {7}{2}}\ln ^{2}\!2-(3\gamma -2\pi )\ln 2+{\frac {\gamma \pi }{2}}+\gamma _{1}\,=\,-0,391298902404\ldots \!\,,}$ (OEIS A254348),

${\displaystyle \displaystyle \gamma _{1}\!\left(\!{\frac {1}{\,3\,}}\!\right)=\,-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}\ln ^{2}\!3+{\frac {\pi }{4{\sqrt {3\,}}}}\left\{\ln 3-8\ln 2\pi -2\gamma +12\ln \Gamma \!\left(\!{\frac {1}{\,3\,}}\!\right)\!\right\}+\,\gamma _{1}\,=\,-3,259557515917\ldots \!\,,}$ (OEIS A254331).

Vrednosti v točkah 2/3, 1/6 in 5/6 je izračunal Blagouchine:^[24]

${\displaystyle \displaystyle \gamma _{1}\!\left(\!{\frac {2}{\,3\,}}\!\right)=\,-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}\ln ^{2}\!3-{\frac {\pi }{4{\sqrt {3\,}}}}\left\{\ln 3-8\ln 2\pi -2\gamma +12\ln \Gamma \!\left(\!{\frac {1}{\,3\,}}\!\right)\!\right\}+\,\gamma _{1}\,=\,-0,5989062842859\ldots \!\,,}$ (OEIS A254345),

${\displaystyle {\begin{aligned}\displaystyle \gamma _{1}\!\left(\!{\frac {1}{\,6\,}}\!\right)=&-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}\ln ^{2}\!3-\ln ^{2}\!2-(3\ln 3+2\gamma )\ln 2+{\frac {3\pi {\sqrt {3\,}}}{2}}\ln \Gamma \!\left(\!{\frac {1}{\,6\,}}\!\right)\\\displaystyle &-{\frac {\pi }{2{\sqrt {3\,}}}}\left\{3\ln 3+11\ln 2+{\frac {15}{2}}\ln \pi +3\gamma \right\}+\,\gamma _{1}\,=\,-10,742582529547\ldots \!\,,\end{aligned}}}$ (OEIS A254349),

${\displaystyle {\begin{aligned}\displaystyle \gamma _{1}\!\left(\!{\frac {5}{\,6\,}}\!\right)=&-{\frac {3\gamma }{2}}\ln 3-{\frac {3}{4}}\ln ^{2}\!3-\ln ^{2}\!2-(3\ln 3+2\gamma )\ln 2-{\frac {3\pi {\sqrt {3\,}}}{2}}\ln \Gamma \!\left(\!{\frac {1}{\,6\,}}\!\right)\\\displaystyle &+{\frac {\pi }{2{\sqrt {3\,}}}}\left\{3\ln 3+11\ln 2+{\frac {15}{2}}\ln \pi +3\gamma \right\}+\,\gamma _{1}\,=\,-0,246169003811\ldots \!\,,\end{aligned}}}$ (OEIS A254350),

Podal je tudi vrednosti v točkah 1/5, 1/8 in 1/12:

${\displaystyle {\begin{aligned}\displaystyle \gamma _{1}{\biggl (}\!{\frac {1}{5}}\!{\biggr )}=&\displaystyle \,\,\,\gamma _{1}+{\frac {\sqrt {5}}{2}}\!\left\{\zeta ''\!\left(\!0,\,{\frac {1}{5}}\!\right)+\zeta ''\!\left(\!0,\,{\frac {4}{5}}\!\right)\!\right\}+{\frac {\pi {\sqrt {10+2{\sqrt {5}}}}}{2}}\ln \Gamma {\biggl (}\!{\frac {1}{5}}\!{\biggr )}\\[5mm]&\displaystyle +{\frac {\pi {\sqrt {10-2{\sqrt {5}}}}}{2}}\ln \Gamma {\biggl (}\!{\frac {2}{5}}\!{\biggr )}+\left\{\!{\frac {\sqrt {5}}{2}}\ln {2}-{\frac {\sqrt {5}}{2}}\ln \!{\big (}1+{\sqrt {5}}{\big )}-{\frac {5}{4}}\ln 5-{\frac {\pi {\sqrt {25+10{\sqrt {5}}}}}{10}}\right\}\!\cdot \gamma \\[5mm]&\displaystyle -{\frac {\sqrt {5}}{2}}\left\{\ln 2+\ln 5+\ln \pi +{\frac {\pi {\sqrt {25-10{\sqrt {5}}}}}{10}}\right\}\!\cdot \ln \!{\big (}1+{\sqrt {5}})+{\frac {\sqrt {5}}{2}}\ln ^{2}\!2+{\frac {{\sqrt {5}}{\big (}1-{\sqrt {5}}{\big )}}{8}}\ln ^{2}\!5\\[5mm]&\displaystyle +{\frac {3{\sqrt {5}}}{4}}\ln 2\cdot \ln 5+{\frac {\sqrt {5}}{2}}\ln 2\cdot \ln \pi +{\frac {\sqrt {5}}{4}}\ln 5\cdot \ln \pi -{\frac {\pi {\big (}2{\sqrt {25+10{\sqrt {5}}}}+5{\sqrt {25+2{\sqrt {5}}}}{\big )}}{20}}\ln 2\\[5mm]&\displaystyle -{\frac {\pi {\big (}4{\sqrt {25+10{\sqrt {5}}}}-5{\sqrt {5+2{\sqrt {5}}}}{\big )}}{40}}\ln 5-{\frac {\pi {\big (}5{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {25+10{\sqrt {5}}}}{\big )}}{10}}\ln \pi \\[5mm]&\displaystyle =-8,030205511035\ldots \!\,,\end{aligned}}}$ (OEIS A251866),

${\displaystyle {\begin{aligned}\displaystyle \gamma _{1}{\biggl (}\!{\frac {1}{8}}\!{\biggr )}=&\displaystyle \,\,\,\gamma _{1}+{\sqrt {2}}\left\{\zeta ''\!\left(\!0,\,{\frac {1}{8}}\!\right)+\zeta ''\!\left(\!0,\,{\frac {7}{8}}\right)\!\right\}+2\pi {\sqrt {2}}\ln \Gamma {\biggl (}\!{\frac {1}{8}}\!{\biggr )}-\pi {\sqrt {2}}{\big (}1-{\sqrt {2}}{\big )}\ln \Gamma {\biggl (}\!{\frac {1}{4}}\!{\biggr )}\\[5mm]&\displaystyle -\left\{\!{\frac {1+{\sqrt {2}}}{2}}\pi +4\ln {2}+{\sqrt {2}}\ln \!{\big (}1+{\sqrt {2}}{\big )}\!\right\}\!\cdot \gamma -{\frac {1}{\sqrt {2}}}{\big (}\pi +8\ln 2+2\ln \pi {\big )}\!\cdot \ln \!{\big (}1+{\sqrt {2}})\\[5mm]&\displaystyle -{\frac {7{\big (}4-{\sqrt {2}}{\big )}}{4}}\ln ^{2}\!2+{\frac {1}{\sqrt {2}}}\ln 2\cdot \ln \pi -{\frac {\pi {\big (}10+11{\sqrt {2}}{\big )}}{4}}\ln 2-{\frac {\pi {\big (}3+2{\sqrt {2}}{\big )}}{2}}\ln \pi \\[5mm]&\displaystyle =-16,641719763609\ldots \end{aligned}}}$ (OEIS A255188),

${\displaystyle {\begin{aligned}\displaystyle \gamma _{1}{\biggl (}\!{\frac {1}{12}}\!{\biggr )}=&\displaystyle \,\,\,\gamma _{1}+{\sqrt {3}}\left\{\zeta ''\!\left(\!0,\,{\frac {1}{12}}\!\right)+\zeta ''\!\left(\!0,\,{\frac {11}{12}}\right)\!\right\}+4\pi \ln \Gamma {\biggl (}\!{\frac {1}{4}}\!{\biggr )}+3\pi {\sqrt {3}}\ln \Gamma {\biggl (}\!{\frac {1}{3}}\!{\biggr )}\\[5mm]&\displaystyle -\left\{\!{\frac {2+{\sqrt {3}}}{2}}\pi +{\frac {3}{2}}\ln 3-{\sqrt {3}}(1-{\sqrt {3}})\ln {2}+2{\sqrt {3}}\ln \!{\big (}1+{\sqrt {3}}{\big )}\!\right\}\!\cdot \gamma \\[5mm]&\displaystyle -2{\sqrt {3}}{\big (}3\ln 2+\ln 3+\ln \pi {\big )}\!\cdot \ln \!{\big (}1+{\sqrt {3}})-{\frac {7-6{\sqrt {3}}}{2}}\ln ^{2}\!2-{\frac {3}{4}}\ln ^{2}\!3\\[5mm]&\displaystyle +{\frac {3{\sqrt {3}}(1-{\sqrt {3}})}{2}}\ln 3\cdot \ln 2+{\sqrt {3}}\ln 2\cdot \ln \pi -{\frac {\pi {\big (}17+8{\sqrt {3}}{\big )}}{2{\sqrt {3}}}}\ln 2\\[5mm]&\displaystyle +{\frac {\pi {\big (}1-{\sqrt {3}}{\big )}{\sqrt {3}}}{4}}\ln 3-\pi {\sqrt {3}}(2+{\sqrt {3}})\ln \pi =-29,842878232041\ldots \!\,,\end{aligned}}}$ (OEIS A255189),

kakor tudi nekatere druge vrednosti.

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:

${\displaystyle {\begin{array}{rl}\displaystyle \gamma _{2}{\biggl (}{\frac {r}{m}}{\biggr )}=\,\gamma _{2}+{\frac {2}{3}}\!\sum _{l=1}^{m-1}\cos {\frac {2\pi rl}{m}}\cdot \zeta '''\!\left(\!0,\,{\frac {l}{m}}\!\right)-2(\gamma +\ln 2\pi m)\!\sum _{l=1}^{m-1}\cos {\frac {2\pi rl}{m}}\cdot \zeta ''\!\left(\!0,\,{\frac {l}{m}}\!\right)\\[6mm]\displaystyle \quad +\pi \!\sum _{l=1}^{m-1}\sin {\frac {2\pi rl}{m}}\cdot \zeta ''\!\left(\!0,\,{\frac {l}{m}}\!\right)-2\pi (\gamma +\ln 2\pi m)\!\sum _{l=1}^{m-1}\sin {\frac {2\pi rl}{m}}\cdot \ln \Gamma {\biggl (}{\frac {l}{m}}{\biggr )}-2\gamma _{1}\ln {m}\\[6mm]\displaystyle \quad -\gamma ^{3}-\left[(\gamma +\ln 2\pi m)^{2}-{\frac {\pi ^{2}}{12}}\right]\!\cdot \!\digamma \!{\biggl (}{\frac {r}{m}}{\biggr )}+{\frac {\pi ^{3}}{12}}\cot {\frac {\pi r}{m}}-\gamma ^{2}\ln {\big (}4\pi ^{2}m^{3}{\big )}+{\frac {\pi ^{2}}{12}}(\gamma +\ln {m})\\[6mm]\displaystyle \quad -\gamma {\big (}\ln ^{2}\!{2\pi }+4\ln {m}\cdot \ln {2\pi }+2\ln ^{2}\!{m}{\big )}-\left\{\!\ln ^{2}\!{2\pi }+2\ln {2\pi }\cdot \ln {m}+{\frac {2}{3}}\ln ^{2}\!{m}\!\right\}\!\ln {m}\end{array}}\,,\qquad \quad (r=1,2,3,\ldots ,m-1)\!\,.}$

Podobni rezultat je kasneje dobil Coffey z drugo metodo.^[27]

• Weisstein, Eric Wolfgang. »Stieltjes Constants«. MathWorld.