Constante de Stieltjes

Constante de Stieltjes
Simbol	$\gamma _{n}$ ${\ Displaystyle \ gamma _ {n}}$ $\ Gamma _ {n}$
Originea numelui	matematician de la care ia constanta numele acestuia
Camp	numere reale
constante corelate	Euler-Mascheroni constanta

Aria converge regiunii albastre la constanta Euler-Mascheroni , care este 0-lea constantă a Stieltjes.

În matematică , Stieltjes constantele $\gamma _{n}$ ${\ Displaystyle \ gamma _ {n}}$ $\ Gamma _ {n}$ sunt coeficienții care apar în seria Laurent extinderea funcției zeta Riemann :

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

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

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

Constanta $\gamma _{0}=\gamma =0,577\ldots$ ${\ Displaystyle \ gamma _ {0} = \ gamma = 0.577 \ ldots}$ ${\ displaystyle \ gamma _ {0} = \ gamma = 0.577 \ ldots}$ este cel mai bine cunoscut ca fiind constanta Euler-Mascheroni .

Reprezentări

Constantele Stieltjes sunt date de limita

\gamma _{n}=\lim _{m\rightarrow \infty }{\left\{\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}-{\frac {(\ln m)^{n+1}}{n+1}}\right\}},

{\ Displaystyle \ gamma _ {n} = \ lim _ {m \ rightarrow \ infty} {\ stângă \ {\ sum _ {k = 1} ^ {m} {\ frac {(\ ln k) ^ {n} } {k}} - {\ frac {(\ ln m) ^ {n + 1}} {n + 1}} \ dreapta \}}}

{\ Displaystyle \ gamma _ {n} = \ lim _ {m \ rightarrow \ infty} {\ stângă \ {\ sum _ {k = 1} ^ {m} {\ frac {(\ ln k) ^ {n} } {k}} - {\ frac {(\ ln m) ^ {n + 1}} {n + 1}} \ dreapta \}}}

in caz $n=0$ ${\ displaystyle n = 0}$ $n = 0$ , În primele apare însumare $0^{0}$ ${\ displaystyle 0 ^ {0}}$ ${\ displaystyle 0 ^ {0}}$ , Care este egal cu 1.

Formula lui Cauchy oferă o reprezentare integrală

\gamma _{n}={\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)dx.

{\ Displaystyle \ gamma _ {n} = {\ frac {(-1) ^ {n} n} {2 \ pi}} \ int _ {0} ^ {2 \ pi} e ^ {- nix} \ zeta \ stânga (e ^ {ix} +1 \ dreapta) dx.}

{\ Displaystyle \ gamma _ {n} = {\ frac {(-1) ^ {n} n} {2 \ pi}} \ int _ {0} ^ {2 \ pi} e ^ {- nix} \ zeta \ stânga (e ^ {ix} +1 \ dreapta) dx.}

Alte reprezentări în ceea ce privește seria și integrală apar în lucrările lui Jensen , Franel, Hermite , Hardy , Ramanujan , Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine și alți autori. ^[1] ^[2] ^[3] ^[4] ^[5] ^[6] în particular, integral formula de Jensen-Franel, adesea atribuit în mod eronat Ainsworth și Howell, afirmă că

\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

{\ 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}} \ dreapta \} \ ,, \ prototipurilor \ quad n = 0,1,2, \ ldots}

{\ 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}} \ dreapta \} \ ,, \ prototipurilor \ quad n = 0,1,2, \ ldots}

unde este $\delta _{n,k}$ ${\ Displaystyle \ delta _ {n, k}}$ ${\ Displaystyle \ delta _ {n, k}}$ este delta Kronecker . ^[5] ^[6] Mai mult decât atât, avem următoarele identități ^[1] ^[5] ^[7]

\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

{\ Displaystyle \ gamma _ {n} = - {\ frac {\ pi} {2 (n + 1)}} \ int _ {- \ infty} ^ {\ infty} {\ frac {\ stânga (\ ln \ stânga ({\ frac {1} {2}} \ pm ix \ dreapta) \ dreapta) ^ {n + 1}} {\ cosh ^ {2} \ pi x}} \ dx \ prototipurilor n = 0,1 , 2, \ ldots}

{\ Displaystyle \ gamma _ {n} = - {\ frac {\ pi} {2 (n + 1)}} \ int _ {- \ infty} ^ {\ infty} {\ frac {\ stânga (\ ln \ stânga ({\ frac {1} {2}} \ pm ix \ dreapta) \ dreapta) ^ {n + 1}} {\ cosh ^ {2} \ pi x}} \ dx \ prototipurilor n = 0,1 , 2, \ ldots}

{\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}}

{\ Displaystyle {\ begin {array} {l} \ displaystyle \ gamma _ {1} = - \ stânga [\ 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}} \ dreapta \} \\ [6mm] \ displaystyle \ gamma _ {1} = - \ gamma ^ {2} - \ int _ {0} ^ {\ infty} \ stânga [{\ frac {1} {1-e ^ {- x}}} - {\ frac {1} {x}} \ right] e ^ {- x} \ ln x \, dx. \ end {array}}}

{\ Displaystyle {\ begin {array} {l} \ displaystyle \ gamma _ {1} = - \ stânga [\ 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}} \ dreapta \} \\ [6mm] \ displaystyle \ gamma _ {1} = - \ gamma ^ {2} - \ int _ {0} ^ {\ infty} \ stânga [{\ frac {1} {1-e ^ {- x}}} - {\ frac {1} {x}} \ right] e ^ {- x} \ ln x \, dx. \ end {array}}}

În ceea ce privește reprezentările serie, în 1912 Hardy ^[8] a constatat următoarele serii , în care partea întreagă apare o carte de logaritmi,

\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).

{\ Displaystyle \ gamma _ {1} = {\ frac {\ ln 2} {2}} \ sum _ {k = 2} ^ {\ infty} {\ frac {(-1) ^ {k}} {k }} \ lfloor \ log _ {2} {k} \ rfloor \ cdot \ stânga (2 \ log _ {2} {k} - \ lfloor \ log _ {2} {2k} \ rfloor \ dreapta)}.

{\ Displaystyle \ gamma _ {1} = {\ frac {\ ln 2} {2}} \ sum _ {k = 2} ^ {\ infty} {\ frac {(-1) ^ {k}} {k }} \ lfloor \ log _ {2} {k} \ rfloor \ cdot \ stânga (2 \ log _ {2} {k} - \ lfloor \ log _ {2} {2k} \ rfloor \ dreapta)}.

Israilov ^[9] a descoperit o serie de semi-convergente în ceea ce privește numerele Bernoulli $B_{2k}$ ${\ B_ displaystyle {2k}}$ ${\ B_ displaystyle {2k}}$

\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

{\ 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)!}} \ stânga [{\ frac {(\ ln x) ^ {m}} {x}} \ right] _ {x = n} ^ {(-2k 1) } - \ theta \ cdot {\ frac {B_ {2N}} {! (2N)}} \ stânga [{\ frac {(\ ln x) ^ {m}} {x}} \ right] _ {x = n} ^ {(2N-1)} \ ,, \ prototipurilor 0 <\ theta <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)!}} \ stânga [{\ frac {(\ ln x) ^ {m}} {x}} \ right] _ {x = n} ^ {(-2k 1) } - \ theta \ cdot {\ frac {B_ {2N}} {! (2N)}} \ stânga [{\ frac {(\ ln x) ^ {m}} {x}} \ right] _ {x = n} ^ {(2N-1)} \ ,, \ prototipurilor 0 <\ theta <1}

Connon, ^[10] Blagouchine ^[6] și Coppo ^[1] în schimb furnizate mai multe serii implicând coeficienți binomice

{\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}}

{\ 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} \ dreapta | \ sum _ {k = 0} ^ {n} (- 1) ^ {k} {\ binom {n} {k}} {\ frac {(\ ln (k + 1)) ^ {m}} { k + 1}} \ end {array}}}

{\ 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} \ dreapta | \ sum _ {k = 0} ^ {n} (- 1) ^ {k} {\ binom {n} {k}} {\ frac {(\ ln (k + 1)) ^ {m}} { k + 1}} \ end {array}}}

unde este $G_{n}$ ${\ G_ displaystyle {n}}$ $G_ {n}$ sunt coeficienții Gregory , de asemenea , cunoscut sub numele de numere logaritmice reciproce ( $G_{1}=+1/2$ ${\ G_ displaystyle {1} = + 1/2}$ ${\ G_ displaystyle {1} = + 1/2}$ , $G_{2}=-1/12$ ${\ G_ displaystyle {2} = - 1/12}$ ${\ G_ displaystyle {2} = - 1/12}$ , $G_{3}=+1/24$ ${\ G_ displaystyle {3} = + 1/24}$ ${\ G_ displaystyle {3} = + 1/24}$ , $G_{4}=-19/720$ ${\ G_ displaystyle {4} = - 19/720}$ ${\ G_ displaystyle {4} = - 19/720}$ , ...). Oloa și Tauraso ^[11] a arătat că anumite serii cu numere armonice duce la constantele Stieltjes

{\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}}

{\ 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}}}

{\ 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 ^[6] a obținut o serie converge lent în care numerele Stirling de primul tip unsigned apar $\left[{\cdot \atop \cdot }\right]$ ${\ Displaystyle \ stânga [{\ cdot \ vârful \ cdot} \ right]}$ ${\ Displaystyle \ stânga [{\ cdot \ vârful \ cdot} \ right]}$

\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,...,

{\ 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 \ stânga [{2k + 2 \ vârful m + 1} \ dreapta] \ cdot \ stânga [{n \ vârful 2k + 1} \ right]} {(2 \ pi) ^ {2k +1}}} \ ,, \ prototipurilor 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 \ stânga [{2k + 2 \ vârful m + 1} \ dreapta] \ cdot \ stânga [{n \ vârful 2k + 1} \ right]} {(2 \ pi) ^ {2k +1}}} \ ,, \ prototipurilor m = 0,1,2, ...,}

împreună cu o serie de semi-convergente cu termeni raționali numai

\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,

{\ Displaystyle \ gamma _ {m} = {\ frac {1} {2}} \ delta _ {m, 0} +! (- 1) ^ {m} m \ cdot \ sum _ {k = 1} ^ {N} {\ frac {\ stânga [{2k \ vârful m + 1} \ right] \ cdot B_ {2k}} {(2k)!}} + \ theta \ cdot {\ frac {(-1) ^ { m} m! \ cdot \ stânga [{2N + 2 \ vârful m + 1} \ right] \ cdot B_ {2N + 2}} {(2N + 2)!}} \ prototipurilor 0 <\ theta <1, }

{\ Displaystyle \ gamma _ {m} = {\ frac {1} {2}} \ delta _ {m, 0} +! (- 1) ^ {m} m \ cdot \ sum _ {k = 1} ^ {N} {\ frac {\ stânga [{2k \ vârful m + 1} \ right] \ cdot B_ {2k}} {(2k)!}} + \ theta \ cdot {\ frac {(-1) ^ { m} m! \ cdot \ stânga [{2N + 2 \ vârful m + 1} \ right] \ cdot B_ {2N + 2}} {(2N + 2)!}} \ prototipurilor 0 <\ theta <1, }

unde este $M=0,1,2,\ldots$ ${\ Displaystyle M = 0,1,2, \ ldots}$ ${\ Displaystyle M = 0,1,2, \ ldots}$ . În special, seria de primă constantă Stieltjes' are o formă extrem de simplu

\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,

{\ 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}}, \ prototipurilor 0 <\ theta <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}}, \ prototipurilor 0 <\ theta <1}

unde este $H_{n}$ ${\ Displaystyle H_ {n}}$ $H_ {n}$ este $n$ ${\ displaystyle n}$ $n$ -lea număr armonic . ^[6] serii mult mai complicate sunt furnizate în scrierile lui Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-Tu, Williams, Coffey. ^[2] ^[3] ^[6]

Estimările și tendința asimptotică

Constantele Stieltjes în valoare absolută îndeplinesc următoarele majorant

|\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}}

{\ Displaystyle | \ gamma _ {n} |! \ Leq {\ begin {cazuri} \ displaystyle {\ frac {2 (n-1)} {\ pi ^ {n}}} \ ,, \ prototipurilor & n = 1, 3,5, \ ldots \\ [3mm] \ displaystyle {\ frac {4 (n-1)!} {\ Pi ^ {n}}} \ ,, \ prototipurilor & n = 2,4,6, \ ldots \ end {cazuri}}}

{\ Displaystyle | \ gamma _ {n} |! \ Leq {\ begin {cazuri} \ displaystyle {\ frac {2 (n-1)} {\ pi ^ {n}}} \ ,, \ prototipurilor & n = 1, 3,5, \ ldots \\ [3mm] \ displaystyle {\ frac {4 (n-1)!} {\ Pi ^ {n}}} \ ,, \ prototipurilor & n = 2,4,6, \ ldots \ end {cazuri}}}

descoperit de Berndt in 1972. ^[12] Cele mai bune estimări în ceea ce privește funcțiile elementare au fost obținute prin Lavrik ^[13]

|\gamma _{n}|\leq {\frac {n!}{2^{n+1}}},\qquad n=1,2,3,\ldots

{\ Displaystyle | \ gamma _ {n} |! \ Leq {\ frac {n} {2 ^ {n + 1}}}, \ n prototipurilor ldots = 1,2,3, \}

{\ Displaystyle | \ gamma _ {n} |! \ Leq {\ frac {n} {2 ^ {n + 1}}}, \ n prototipurilor ldots = 1,2,3, \}

și prin Israilov ^[9]

|\gamma _{n}|\leq {\frac {n!C(k)}{(2k)^{n}}},\qquad n=1,2,3,\ldots

{\ Displaystyle | \ gamma _ {n} |! \ Leq {\ frac {n C (k)} {(2k) ^ {n}}}, \ n prototipurilor ldots = 1,2,3, \}

{\ Displaystyle | \ gamma _ {n} |! \ Leq {\ frac {n C (k)} {(2k) ^ {n}}}, \ n prototipurilor ldots = 1,2,3, \}

cu $k=1,2,\ldots$ ${\ Displaystyle k = 1,2, \ ldots}$ ${\ displaystyle k = 1,2, \ ldots}$ Și $C(1)=1/2$ ${\ Displaystyle C (1) = 1/2}$ ${\ Displaystyle C (1) = 1/2}$ , $C(2)=7/12$ ${\ Displaystyle C (2) = 7/12}$ ${\ Displaystyle C (2) = 7/12}$ , ...; Nan-Tu și Williams ^[14]

|\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}}

{\ Displaystyle | \ gamma _ {n} | \ leq {\ begin {cazuri} \ displaystyle {\ frac {2 (2n)} {n ^ {n + 1} (2 \ pi) ^ {n}}}! \ ,, \ prototipurilor & n = 1,3,5, \ ldots \\ [4mm] \ displaystyle {\ frac {4 (2n)!} {n ^ {n + 1} (2 \ pi) ^ {n} }} \ ,, \ prototipurilor & n = 2,4,6, \ ldots \ end {cazuri}}}

{\ Displaystyle | \ gamma _ {n} | \ leq {\ begin {cazuri} \ displaystyle {\ frac {2 (2n)} {n ^ {n + 1} (2 \ pi) ^ {n}}}! \ ,, \ prototipurilor & n = 1,3,5, \ ldots \\ [4mm] \ displaystyle {\ frac {4 (2n)!} {n ^ {n + 1} (2 \ pi) ^ {n} }} \ ,, \ prototipurilor & n = 2,4,6, \ ldots \ end {cazuri}}}

și , de asemenea , prin Blagouchine ^[6]

{\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}}

{\ Displaystyle {\ begin {array} {ll} \ displaystyle - {\ frac {{\ big |} {B} _ {m + 1} {\ mare |}} {m + 1}} <\ gamma _ { m} <{\ frac {(3m + 8) \ cdot {\ mare |} {B} _ {m + 3} {\ mare |}} {24}} - {\ frac {{\ big |} {B } _ {m + 1} {\ big |}} {m + 1}}, & m = 1,5,9, \ ldots \\ [12pt] \ displaystyle {\ frac {{\ mare |} B_ {m + 1} {\ mare |}} {m + 1}} - {\ frac {(3m + 8) \ cdot {\ mare |} B_ {m + 3} {\ mare |}} {24}} <\ gamma _ {m} <{\ frac {{\ mare |} {B} _ {m + 1} {\ big |}} {m + 1}}, & m = 3,7,11, \ ldots \\ [12pt] \ displaystyle - {\ frac {{\ big |} {B} _ {m + 2} {\ mare |}} {2}} <\ gamma _ {m} <{\ frac {(m + 3 ) (m + 4) \ cdot {\ mare |} {B} _ {m + 4} {\ mare |}} {48}} - {\ frac {{\ mare |} B_ {m + 2} {\ mare |}} {2}}, \ prototipurilor & m = 2,6,10, \ ldots \\ [12pt] \ displaystyle {\ frac {{\ big |} {B} _ {m + 2} {\ mare |}} {2}} - {\ frac {(m + 3) (m + 4) \ cdot {\ mare |} {B} _ {m + 4} {\ mare |}} {48}} <\ gamma _ {m} <{\ frac {{\ big |} {} _ {m + 2} {\ big B |}} {2}}, & m = 4,8,12, \ end ldots \\\ {array}}}

{\ Displaystyle {\ begin {array} {ll} \ displaystyle - {\ frac {{\ big |} {B} _ {m + 1} {\ mare |}} {m + 1}} <\ gamma _ { m} <{\ frac {(3m + 8) \ cdot {\ mare |} {B} _ {m + 3} {\ mare |}} {24}} - {\ frac {{\ big |} {B } _ {m + 1} {\ big |}} {m + 1}}, & m = 1,5,9, \ ldots \\ [12pt] \ displaystyle {\ frac {{\ mare |} B_ {m + 1} {\ mare |}} {m + 1}} - {\ frac {(3m + 8) \ cdot {\ mare |} B_ {m + 3} {\ mare |}} {24}} <\ gamma _ {m} <{\ frac {{\ mare |} {B} _ {m + 1} {\ big |}} {m + 1}}, & m = 3,7,11, \ ldots \\ [12pt] \ displaystyle - {\ frac {{\ big |} {B} _ {m + 2} {\ mare |}} {2}} <\ gamma _ {m} <{\ frac {(m + 3 ) (m + 4) \ cdot {\ mare |} {B} _ {m + 4} {\ mare |}} {48}} - {\ frac {{\ mare |} B_ {m + 2} {\ mare |}} {2}}, \ prototipurilor & m = 2,6,10, \ ldots \\ [12pt] \ displaystyle {\ frac {{\ big |} {B} _ {m + 2} {\ mare |}} {2}} - {\ frac {(m + 3) (m + 4) \ cdot {\ mare |} {B} _ {m + 4} {\ mare |}} {48}} <\ gamma _ {m} <{\ frac {{\ big |} {} _ {m + 2} {\ big B |}} {2}}, & m = 4,8,12, \ end ldots \\\ {array}}}

unde este $B_{n}$ ${\ displaystyle B_ {n}}$ $B_ {n}$ sunt numerele Bernoulli . În cele din urmă , avem următoarea estimare a Matsuoka ^[15] ^[16]

|\gamma _{n}|<10^{-4}e^{n\ln \ln n}\,,\qquad n=5,6,7,\ldots

{\ Displaystyle | \ gamma _ {n} | <10 ^ {- 4} e ^ {n \ ln \ ln n} \ ,, \ prototipurilor n = 5,6,7, \ ldots}

{\ Displaystyle | \ gamma _ {n} | <10 ^ {- 4} e ^ {n \ ln \ ln n} \ ,, \ prototipurilor n = 5,6,7, \ ldots}

În ceea ce privește estimările prin intermediul funcțiilor non-elementare și soluțiile lor, Knessl, Coffey ^[17] și Fekih-Ahmed ^[18] au obținut rezultate destul de precise. De exemplu, Knessl și Coffey furnizat următoarea formulă care aproximează constantele Stieltjes relativ bine $n$ ${\ displaystyle n}$ $n$ Grozav. ^[17] Dacă $v$ ${\ displaystyle v}$ $v$ este soluția unică

2\pi \exp(v\tan v)=n{\frac {\cos(v)}{v}}

{\ Displaystyle 2 \ pi \ exp (v \ v tan) = n {\ frac {\ cos (v)} {v}}}

{\ Displaystyle 2 \ pi \ exp (v \ v tan) = n {\ frac {\ cos (v)} {v}}}

cu $0<v<\pi /2$ ${\ Displaystyle 0 <v <\ pi / 2}$ ${\ Displaystyle 0 <v <\ pi / 2}$ , si daca $u=v\tan v$ ${\ Displaystyle u = v \ v tan}$ ${\ Displaystyle u = v \ v tan}$ , asa de

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

{\ Displaystyle \ gamma _ {n} \ sim {\ frac {B} {\ sqrt {n}}} {e ^ nA} \ cos (un + b)}

{\ Displaystyle \ gamma _ {n} \ sim {\ frac {B} {\ sqrt {n}}} {e ^ nA} \ cos (un + b)}

unde este

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

{\ Displaystyle A = {\ frac {1} {2}} \ ln (u ^ {2} + v ^ {2}) - {\ frac {u} {u ^ {2} + v ^ {2}} }}

{\ Displaystyle A = {\ frac {1} {2}} \ ln (u ^ {2} + v ^ {2}) - {\ frac {u} {u ^ {2} + v ^ {2}} }}

B={\frac {2{\sqrt {2\pi }}{\sqrt {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}}}}

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

a=\tan ^{-1}\left({\frac {v}{u}}\right)+{\frac {v}{u^{2}+v^{2}}}

{\ Displaystyle a = \ tan ^ {- 1} \ stânga ({\ frac {v} {u}} \ dreapta) + {\ frac {v} {u ^ {2} + v ^ {2}}}}

{\ Displaystyle a = \ tan ^ {- 1} \ stânga ({\ frac {v} {u}} \ dreapta) + {\ frac {v} {u ^ {2} + v ^ {2}}}}

b=\tan ^{-1}\left({\frac {v}{u}}\right)-{\frac {1}{2}}\left({\frac {v}{u+1}}\right).

{\ Displaystyle b = \ tan ^ {- 1} \ stânga ({\ frac {v} {u}} \ dreapta) - {\ frac {1} {2}} \ stânga ({\ frac {v} {u +1}} \ dreapta).}

{\ Displaystyle b = \ tan ^ {- 1} \ stânga ({\ frac {v} {u}} \ dreapta) - {\ frac {1} {2}} \ stânga ({\ frac {v} {u +1}} \ dreapta).}

Până la $n=100000$ ${\ N displaystyle = 100000}$ ${\ N displaystyle = 100000}$ , Apropierea Knessl-Coffey a prezis corect semnul $\gamma _{n}$ ${\ Displaystyle \ gamma _ {n}}$ $\ Gamma _ {n}$ , Cu singura excepție $n=137$ ${\ N displaystyle = 137}$ ${\ N displaystyle = 137}$ . ^[17]

valorile numerice

Primele valori ale $\gamma _{n}$ ${\ Displaystyle \ gamma _ {n}}$ $\ Gamma _ {n}$ Sunt:

$n$ ${\ displaystyle n}$ $n$	Valoarea aproximativă $\gamma _{n}$ ${\ Displaystyle \ gamma _ {n}}$ $\ Gamma _ {n}$	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 × 10 ¹⁷
1000	-1,5709538442047449345494023425120825242380299554 × 10 ⁴⁸⁶
10000	-2,2104970567221060862971082857536501900234397174 × 10 ⁶⁸⁸³
100000	+1,9919273063125410956582272431568589205211659777 × 10 ⁸³⁴³²

Pentru $n$ ${\ displaystyle n}$ $n$ mare, constantele Stieltjes cresc rapid în valoare absolută, și semn de schimbare cu un model foarte complex.

Informații suplimentare privind evaluarea numerică a constantelor Stieltjes pot fi găsite în lucrările lui Keiper, ^[19] Kreminski, ^[20] Plouffe ^[21] și Johansson. ^[22] Ultimul autor a dat valorile Stieltjes constante de până la $n=100000$ ${\ N displaystyle = 100000}$ ${\ N displaystyle = 100000}$ , Fiecare la cifra specifică numarul 10.000. Valorile numerice pot fi găsite în LMFDB [1] .

Constantele Stieltjes generalizate

Informații generale

Mai general, constantele Stieltjes pot fi definite $\gamma _{n}(a)$ ${\ Displaystyle \ gamma _ {n} (a)}$ ${\ Displaystyle \ gamma _ {n} (a)}$ care apare în Laurent e serie de funcții zeta Hurwitz lui :

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

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

unde este $la$ ${\ displaystyle a}$ $la$ este un număr complex cu $\operatorname {Re} (a)>0$ ${\ Displaystyle \ operatorname {Re} (a)> 0}$ ${\ Displaystyle \ operatorname {Re} (a)> 0}$ . Deoarece funcția zeta Hurwitz este o generalizare a funcției zeta Riemann, avem că $\gamma _{n}(1)=\gamma _{n}$ ${\ Displaystyle \ gamma _ {n} (1) = \ gamma _ {n}}$ ${\ Displaystyle \ gamma _ {n} (1) = \ gamma _ {n}}$ . Constanta cu $n=0$ ${\ displaystyle n = 0}$ $n = 0$ este pur și simplu funcția digamma $\gamma _{0}(a)=-\psi (a)$ ${\ Displaystyle \ gamma _ {0} (a) = - \ psi (a)}$ ${\ Displaystyle \ gamma _ {0} (a) = - \ psi (a)}$ , ^[23] în timp ce nu se știe dacă celelalte constante pot fi urmărite înapoi la o funcție elementară sau clasică a analizei. Cu toate acestea, există numeroase reprezentări pentru aceste constante. De exemplu, există următoarea reprezentare asimptotică

\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}}

{\ Displaystyle \ gamma _ {n} (a) = \ lim _ {m \ la \ infty} \

din

stânga \ {\ sum _ {k = 0} ^ {m} {\ frac {(\ ln (k + un )) ^ {n}} {+ a}} k - {\ frac {(\ ln (m + a)) ^ {n + 1}} {n + 1}} \ dreapta \} \ prototipurilor {\ începe {array} {l} n = 0,1,2, \ ldots \\ [1mm] a \ neq 0, -1, -2, \ ldots \ end {array}}}

{\ Displaystyle \ gamma _ {n} (a) = \ lim _ {m \ la \ infty} \ din stânga \ {\ sum _ {k = 0} ^ {m} {\ frac {(\ ln (k + un )) ^ {n}} {+ a}} k - {\ frac {(\ ln (m + a)) ^ {n + 1}} {n + 1}} \ dreapta \} \ prototipurilor {\ începe {array} {l} n = 0,1,2, \ ldots \\ [1mm] a \ neq 0, -1, -2, \ ldots \ end {array}}}

ca urmare a Berndt și Wilton. Analogul pentru constantele Stieltjes generalizate cu formula Jensen-Franel este Hermite formula ^[5]

\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}}

{\ Displaystyle \ gamma _ {n} (a) = \ stânga [{\ frac {1} {2a}} - {\ frac {\ ln {a}} {n + 1}} \ right] (\ ln o ) ^ {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}} \ dreapta \} \ prototipurilor {\ începe {array} {l} n = 0,1,2, \ ldots \\ [1mm] a \ neq 0, -1, -2, \ ldots \ end {array}}}

{\ Displaystyle \ gamma _ {n} (a) = \ stânga [{\ frac {1} {2a}} - {\ frac {\ ln {a}} {n + 1}} \ right] (\ ln o ) ^ {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}} \ dreapta \} \ prototipurilor {\ începe {array} {l} n = 0,1,2, \ ldots \\ [1mm] a \ neq 0, -1, -2, \ ldots \ end {array}}}

Constantele Stieltjes generalizate satisfac următoarea relație recursiv

\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}}

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

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

precum teorema de multiplicare

\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

{\ Displaystyle \ sum _ {l = 0} ^ {n-1} \ gamma _ {p} \ stânga (a + {\ frac {l} {n}} \ dreapta) = (- 1) ^ {p} n \ stânga [{\ frac {\ ln n} {p + 1}} - \ Psi (o) \ right] (\ ln n) ^ {p} + n \ sum _ {r = 0} ^ {p- 1} (- 1) ^ {r} {\ binom {p} {r}} \ gamma _ {} pr (o) \ cdot (\ ln n) ^ {r} \ ,, \ prototipurilor \ prototipurilor n = 2 , 3,4, \ ldots}

{\ Displaystyle \ sum _ {l = 0} ^ {n-1} \ gamma _ {p} \ stânga (a + {\ frac {l} {n}} \ dreapta) = (- 1) ^ {p} n \ stânga [{\ frac {\ ln n} {p + 1}} - \ Psi (o) \ right] (\ ln n) ^ {p} + n \ sum _ {r = 0} ^ {p- 1} (- 1) ^ {r} {\ binom {p} {r}} \ gamma _ {} pr (o) \ cdot (\ ln n) ^ {r} \ ,, \ prototipurilor \ prototipurilor n = 2 , 3,4, \ ldots}

unde este ${\binom {p}{r}}$ ${\ Displaystyle {\ binom {p} {r}}}$ ${\ Displaystyle {\ binom {p} {r}}}$ indică coeficientul binomială . ^[24] ^[25]

În primul rând generalizată constantă Stieltjes

Constanta Stieltjes primul generalizată posedă un număr de proprietăți remarcabile.

Identitatea Malmsten (formula de reflexie pentru constanta primul generalizata): formula de reflexie pentru constanta Stieltjes primul generalizată este următoarea

\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}}

{\ Displaystyle \ gamma _ {1} {\ biggl (} {\ frac {m} {n}} {\ biggr)} - \ gamma _ {1} {\ biggl (} 1 - {\ frac {m} { n}} {\ biggr)} = 2 \ pi \ sum _ {l = 1} ^ {n-1} \ păcat {\ frac {2 \ pi ml} {n}} \ cdot \ ln \ Gamma {\ biggl (} {\ frac {l} {n}} {\ biggr)} - \ pi (\ gamma + \ ln 2 \ pi n) \ cot {\ frac {m \ pi} {n}}}

{\ Displaystyle \ gamma _ {1} {\ biggl (} {\ frac {m} {n}} {\ biggr)} - \ gamma _ {1} {\ biggl (} 1 - {\ frac {m} { n}} {\ biggr)} = 2 \ pi \ sum _ {l = 1} ^ {n-1} \ păcat {\ frac {2 \ pi ml} {n}} \ cdot \ ln \ Gamma {\ biggl (} {\ frac {l} {n}} {\ biggr)} - \ pi (\ gamma + \ ln 2 \ pi n) \ cot {\ frac {m \ pi} {n}}}

unde este $m$ ${\ displaystyle m}$ $m$ Și $n$ ${\ displaystyle n}$ $n$ sunt două numere întregi pozitive astfel încât $m<n$ ${\ Displaystyle m <n}$ ${\ Displaystyle m <n}$ . Această formulă a fost mult timp atribuită Almkvist și Meurman care a derivat în anii 1990. ^[26] Cu toate acestea, a fost recent descoperit că identitatea, deși într - o formă ușor diferită, a fost mai întâi obținut prin Carl Malmsten în 1846. ^[5] ^[27]

Rațional argument teorema: constanta Stieltjes primul generalizată poate fi calculată în numere raționale în formă cvasi închisă prin următoarea formulă ^[5] ^[23]

{\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.

{\ 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 \ m pi) \ cdot \ Psi \ stânga ({\ frac {r} {m}} \ dreapta) \\ [5mm] \ displaystyle & \ displaystyle \ prototipurilor + \ pi \ sum _ { l = 1} ^ {m-1} \ păcat {\ 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 '' \ stânga (0, {\ frac {l} {m }} \ dreapta) \ end {array}} \ ,, \ prototipurilor \ quad r = 1,2,3, \ ldots, 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 \ m pi) \ cdot \ Psi \ stânga ({\ frac {r} {m}} \ dreapta) \\ [5mm] \ displaystyle & \ displaystyle \ prototipurilor + \ pi \ sum _ { l = 1} ^ {m-1} \ păcat {\ 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 '' \ stânga (0, {\ frac {l} {m }} \ dreapta) \ end {array}} \ ,, \ prototipurilor \ quad r = 1,2,3, \ ldots, m-1.}

O dovadă alternativă mai târziu a fost propus de Coffey ^[28] și mulți alți autori.

Sume finite: există multe însumări cu privire la constanta Stieltjes prima generalizată. De exemplu:

{\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}}

{\ Displaystyle {\ begin {array} {ll} \ displaystyle \ sum _ {r = 0} ^ {m-1} \ gamma _ {1} \ stânga (a + {\ frac {r} {m}} \ dreapta) = m \ ln {m} \ cdot \ Psi (am) - {\ frac {m} {2}} (\ ln m) ^ {2} + m \ gamma _ {1} (am) \ ,, \ prototipurilor o \ în \ mathbb {C} \\ [6mm] \ displaystyle \ sum _ {r = 1} ^ {m-1} \ gamma _ {1} \ stânga ({\ frac {r} {m}} \ dreapta) = (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 \ stângă \ {4 \ pi \ ln \ Gamma {\ biggl ( } {\ frac {1} {4}} {\ biggr)} - \ pi {\ mare (} 4 \ ln 2 + 3 \ ln \ pi + \ ln m + \ gamma {\ mare)} \ dreapta \} \\ [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 \ stânga (2 \ păcat {\ frac {k \ pi } {m}} \ dreapta) + {\ frac {m} {2}} \ stângă \ {\ zeta '' \ stânga (0, {\ frac {k} {m}} \ dreapta) + \ zeta '' \ stânga (0,1 - {\ frac {k} {m}} \ dreapta) \ drept \} \ ,, \ prototipurilor k = 1,2, \ ld OTS, m-1 \\ [6mm] \ displaystyle \ sum _ {r = 1} ^ {m-1} \ gamma _ {1} {\ biggl (} {\ frac {r} {m}} {\ biggr )} \ cdot \ păcat {\ dfrac {2 \ pi rk} {m}} = {\ frac {\ pi} {2}} (\ gamma + \ ln 2 \ pi m) (2k-m) - {\ frac {\ pi m} {2}} \ stângă \ {\ ln \ pi - \ ln \ păcat {\ frac {k \ pi} {m}} \ dreapta \} + m \ pi \ ln \ Gamma {\ biggl (} {\ frac {k} {m}} {\ biggr)} \ ,, \ prototipurilor k = 1,2, \ ldots, m-1 \\ [6mm] \ displaystyle \ sum _ {r = 1} ^ {m-1} \ gamma _ {1} {\ biggl (} {\ frac {r} {m}} {\ biggr)} \ cdot \ pătuţ {\ 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 \ stânga ({\ frac {l} {m}} \ dreapta) \\ [6mm] \ displaystyle \ sum _ {r = 1} ^ {m-1} {\ frac {r} {m}} \ cdot \ gamma _ {1} {\ biggl ( } {\ frac {r} {m}} {\ biggr)} = {\ frac {1} {2}} \

din

stânga \ {(m-1) \ gamma _ {1} -M \ gamma \ ln {m } - {\ frac {m} {2}} (\ ln m) ^ {2} \ dreapta \} - {\ frac {\ pi} {2m}} (\ gamma + \ ln 2 \ pi m) \ sum _ {l = 1} ^ {m-1} l \ cdot \ pătuţ {\ frac {\ pi l} {m}} - {\ frac {\ pi} {2}} \ sum _ {l = 1} ^ pat copii {m-1} \ {\ frac {\ pi l} {m}} \ cdot \ ln \ Gamma {\ biggl (} {\ frac {l} {m}} {\ biggr)} \ end {array} }}

{\ Displaystyle {\ begin {array} {ll} \ displaystyle \ sum _ {r = 0} ^ {m-1} \ gamma _ {1} \ stânga (a + {\ frac {r} {m}} \ dreapta) = m \ ln {m} \ cdot \ Psi (am) - {\ frac {m} {2}} (\ ln m) ^ {2} + m \ gamma _ {1} (am) \ ,, \ prototipurilor o \ în \ mathbb {C} \\ [6mm] \ displaystyle \ sum _ {r = 1} ^ {m-1} \ gamma _ {1} \ stânga ({\ frac {r} {m}} \ dreapta) = (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 \ stângă \ {4 \ pi \ ln \ Gamma {\ biggl ( } {\ frac {1} {4}} {\ biggr)} - \ pi {\ mare (} 4 \ ln 2 + 3 \ ln \ pi + \ ln m + \ gamma {\ mare)} \ dreapta \} \\ [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 \ stânga (2 \ păcat {\ frac {k \ pi } {m}} \ dreapta) + {\ frac {m} {2}} \ stângă \ {\ zeta '' \ stânga (0, {\ frac {k} {m}} \ dreapta) + \ zeta '' \ stânga (0,1 - {\ frac {k} {m}} \ dreapta) \ drept \} \ ,, \ prototipurilor k = 1,2, \ ld OTS, m-1 \\ [6mm] \ displaystyle \ sum _ {r = 1} ^ {m-1} \ gamma _ {1} {\ biggl (} {\ frac {r} {m}} {\ biggr )} \ cdot \ păcat {\ dfrac {2 \ pi rk} {m}} = {\ frac {\ pi} {2}} (\ gamma + \ ln 2 \ pi m) (2k-m) - {\ frac {\ pi m} {2}} \ stângă \ {\ ln \ pi - \ ln \ păcat {\ frac {k \ pi} {m}} \ dreapta \} + m \ pi \ ln \ Gamma {\ biggl (} {\ frac {k} {m}} {\ biggr)} \ ,, \ prototipurilor k = 1,2, \ ldots, m-1 \\ [6mm] \ displaystyle \ sum _ {r = 1} ^ {m-1} \ gamma _ {1} {\ biggl (} {\ frac {r} {m}} {\ biggr)} \ cdot \ pătuţ {\ 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 \ stânga ({\ frac {l} {m}} \ dreapta) \\ [6mm] \ displaystyle \ sum _ {r = 1} ^ {m-1} {\ frac {r} {m}} \ cdot \ gamma _ {1} {\ biggl ( } {\ frac {r} {m}} {\ biggr)} = {\ frac {1} {2}} \ din stânga \ {(m-1) \ gamma _ {1} -M \ gamma \ ln {m } - {\ frac {m} {2}} (\ ln m) ^ {2} \ dreapta \} - {\ frac {\ pi} {2m}} (\ gamma + \ ln 2 \ pi m) \ sum _ {l = 1} ^ {m-1} l \ cdot \ pătuţ {\ frac {\ pi l} {m}} - {\ frac {\ pi} {2}} \ sum _ {l = 1} ^ pat copii {m-1} \ {\ frac {\ pi l} {m}} \ cdot \ ln \ Gamma {\ biggl (} {\ frac {l} {m}} {\ biggr)} \ end {array} }}

Pentru detalii suplimentare si însumări, vezi ^[5] ^[25] .

Unele valori particulare: Unele valori particulare ale $\gamma _{1}(a)$ ${\ Displaystyle \ gamma _ {1} (a)}$ ${\ Displaystyle \ gamma _ {1} (a)}$ cu $la$ ${\ displaystyle a}$ $la$ rațional pot fi urmărite înapoi la funcția gamma , prima constanta Stieltjes și unele funcții elementare. De exemplu,

\gamma _{1}\left({\frac {1}{2}}\right)=-2\gamma \ln 2-(\ln 2)^{2}+\gamma _{1}=-1,353459680\ldots

\gamma _{1}\left({\frac {1}{2}}\right)=-2\gamma \ln 2-(\ln 2)^{2}+\gamma _{1}=-1,353459680\ldots

\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 $1/4$ ${\displaystyle 1/4}$ $1/4$ , $3/4$ ${\displaystyle 3/4}$ $3/4$ e $1/3$ ${\displaystyle 1/3}$ $1/3$ furono ottenuti indipendentemente da Connon ^[29] e Blagouchine ^[25]

{\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}}

{\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}}

{\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 $2/3$ ${\displaystyle 2/3}$ $2/3$ , $1/6$ ${\displaystyle 1/6}$ $1/6$ e $5/6$ ${\displaystyle 5/6}$ $5/6$

{\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}}

{\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}}

{\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. ^[25] Sono dovute allo stesso autore anche le seguenti identità

{\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}}

{\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}}

{\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

La seconda costante di Stieltjes generalizzata è molto meno studiata della prima. Similmente alla prima, si possono calcolare i valori di $\gamma _{2}(a)$ $\gamma _{2}(a)$ $\gamma _{2}(a)$ con $a$ ${\displaystyle a}$ $a$ razionale e $r=1,2,3,\ldots ,m-1,$ $r=1,2,3,\ldots ,m-1,$ $r=1,2,3,\ldots ,m-1,$ attraverso la seguente formula ^[5]

{\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}}

{\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}}

{\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. ^[28]

Note

^ ^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 Mark W. Coffey. Addison-type series representation for the Stieltjes constants . J. Number Theory, vol. 130, pp. 2049-2064, 2010.
^ 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 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 (Elsevier), vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv PDF
^ ^a ^b ^c ^d ^e ^f ^g Iaroslav V. Blagouchine. 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 (Elsevier), vol. 158, pp. 365-396, 2016. Corrigendum: vol. 173, pp. 631-632, 2017. arXiv:1501.00740
^ Math StackExchange: A couple of definite integrals related to Stieltjes constants
^ GH Hardy. Note on Dr. Vacca's series for γ , QJ Pure Appl. Math. 43, pp. 215–216, 2012.
^ ^a ^b MI 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 StackExchange: A closed form for the series ...
^ Bruce C. Berndt. On the Hurwitz Zeta-function . Rocky Mountain Journal of Mathematics, vol. 2, no. 1, pp. 151-157, 1972.
^ AF 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 KS 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
^ JB 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
^ ^a ^b Math StackExchange: Definite integral
^ Donal F. Connon New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions , arXiv:0903.4539
^ ^a ^b ^c ^d Iaroslav V. Blagouchine Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. Erratum-Addendum: vol. 42, pp. 777-781, 2017. PDF
^ V. Adamchik. A class of logarithmic integrals. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.
^ Math StackExchange: evaluation of a particular integral
^ ^a ^b Mark W. Coffey Functional equations for the Stieltjes constants , arXiv:1402.3746
^ Donal F. Connon The difference between two Stieltjes constants , arXiv:0906.0277

Voci correlate

Collegamenti esterni

( EN ) Eric W. Weisstein,Costanti di Stieltjes , in MathWorld , Wolfram Research.

Portale Matematica : accedi alle voci di Wikipedia che trattano di matematica

[cpo-1] Marc-Antoine Coppo. Nouvelles expressions des constantes de Stieltjes . Expositiones Mathematicae, vol. 17, pp. 349-358, 1999.

[cf1-2] Mark W. Coffey. Series representations for the Stieltjes constants , arXiv:0905.1111

[cf2-3] Mark W. Coffey. Addison-type series representation for the Stieltjes constants . J. Number Theory, vol. 130, pp. 2049-2064, 2010.

[4] Junesang Choi. Certain integral representations of Stieltjes constants , Journal of Inequalities and Applications, 2013:532, pp. 1-10

[blg_02-5] ^ ^a ^b ^c ^d ^e ^f ^g ^h 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 (Elsevier), vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv PDF

[blg_03-6] ^ ^a ^b ^c ^d ^e ^f ^g Iaroslav V. Blagouchine. 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 (Elsevier), vol. 158, pp. 365-396, 2016. Corrigendum: vol. 173, pp. 631-632, 2017. arXiv:1501.00740

[7] Math StackExchange: A couple of definite integrals related to Stieltjes constants

[8] GH Hardy. Note on Dr. Vacca's series for γ , QJ Pure Appl. Math. 43, pp. 215–216, 2012.

[isrl-9] MI Israilov. On the Laurent decomposition of Riemann's zeta function [in Russian] . Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 158, pp. 98-103, 1981.

[10] Donal F. Connon Some applications of the Stieltjes constants , arXiv:0901.2083

[11] Math StackExchange: A closed form for the series ...

[12] Bruce C. Berndt. On the Hurwitz Zeta-function . Rocky Mountain Journal of Mathematics, vol. 2, no. 1, pp. 151-157, 1972.

[13] AF 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.

[14] Z. Nan-You and KS Williams. Some results on the generalized Stieltjes constants . Analysis, vol. 14, pp. 147-162, 1994.

[15] Y. Matsuoka. Generalized Euler constants associated with the Riemann zeta function . Number Theory and Combinatorics: Japan 1984, World Scientific, Singapore, pp. 279-295, 1985

[16] Y. Matsuoka. On the power series coefficients of the Riemann zeta function . Tokyo Journal of Mathematics, vol. 12, no. 1, pp. 49-58, 1989.

[kns-17] Charles Knessl e Mark W. Coffey. An effective asymptotic formula for the Stieltjes constants . Math. Comp., vol. 80, no. 273, pp. 379-386, 2011.

[ahm-18] Lazhar Fekih-Ahmed. A New Effective Asymptotic Formula for the Stieltjes Constants , arXiv:1407.5567

[19] JB Keiper. Power series expansions of Riemann ζ-function . Math. Comp., vol. 58, no. 198, pp. 765-773, 1992.

[20] Rick Kreminski. Newton-Cotes integration for approximating Stieltjes generalized Euler constants . Math. Comp., vol. 72, no. 243, pp. 1379-1397, 2003.

[21] Simon Plouffe. Stieltjes Constants, from 0 to 78, 256 digits each

[22] Fredrik Johansson. Rigorous high-precision computation of the Hurwitz zeta function and its derivatives , arXiv:1309.2877

[mse-23] Math StackExchange: Definite integral

[24] Donal F. Connon New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions , arXiv:0903.4539

[blg_01-25] Iaroslav V. Blagouchine Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. Erratum-Addendum: vol. 42, pp. 777-781, 2017. PDF

[adm-26] V. Adamchik. A class of logarithmic integrals. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.

[27] Math StackExchange: evaluation of a particular integral

[coff-28] Mark W. Coffey Functional equations for the Stieltjes constants , arXiv:1402.3746

[29] Donal F. Connon The difference between two Stieltjes constants , arXiv:0906.0277

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