In mathematics , the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} Laurent series expansion of the Riemann zeta function :
ζ ( 1 + s ) = 1 s + ∑ n = 0 ∞ ( − 1 ) n n ! γ n s n . {\displaystyle \zeta (1+s)={\frac {1}{s}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\gamma _{n}s^{n}.} The constant γ 0 = γ = 0.577 … {\displaystyle \gamma _{0}=\gamma =0.577\dots }
Representations The Stieltjes constants are given by the limit
γ n = lim m → ∞ { ∑ k = 1 m ( ln k ) n k − ∫ 1 m ( ln x ) n x d x } = lim m → ∞ { ∑ k = 1 m ( ln k ) n k − ( ln m ) n + 1 n + 1 } . {\displaystyle \gamma _{n}=\lim _{m\to \infty }\left\{\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}-\int _{1}^{m}{\frac {(\ln x)^{n}}{x}}\,dx\right\}=\lim _{m\rightarrow \infty }{\left\{\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}-{\frac {(\ln m)^{n+1}}{n+1}}\right\}}.} (In the case n = 0, the first summand requires evaluation of 00 , which is taken to be 1.)
Cauchy's differentiation formula leads to the integral representation
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.} Various representations in terms of integrals and infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan , Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors. In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that
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 } where δn,k Kronecker symbol (Kronecker delta) . Among other formulae, we find
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 \qquad \qquad \qquad \qquad \qquad n=0,1,2,\ldots } γ 1 = − [ γ − ln 2 2 ] ln 2 + i ∫ 0 ∞ d x e π x + 1 { ln ( 1 − i x ) 1 − i x − ln ( 1 + i x ) 1 + i x } γ 1 = − γ 2 − ∫ 0 ∞ [ 1 1 − e − x − 1 x ] e − x ln x d x {\displaystyle {\begin{array}{l}\displaystyle \gamma _{1}=-\left[\gamma -{\frac {\ln 2}{2}}\right]\ln 2+i\int _{0}^{\infty }{\frac {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}}} see.
As concerns series representations, a famous series employing an integer part of a logarithm was given by Hardy in 1912
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 gave semi-convergent series in terms of Bernoulli numbers 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, Blagouchine and Coppo gave several series with the binomial coefficients
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 = − 1 m + 1 ∑ n = 0 ∞ H n + 1 ∑ k = 0 n ( − 1 ) k ( n k ) ( ln ( k + 2 ) ) m + 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}=-{\frac {1}{m+1}}\sum _{n=0}^{\infty }H_{n+1}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(\ln(k+2))^{m+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}}} where G n G 1 =+1/2, G 2 =−1/12, G 3 =+1/24, G 4 =−19/720,... ). More general series of the same nature include these examples
γ m = − ( ln ( 1 + a ) ) m + 1 m + 1 + ∑ n = 0 ∞ ( − 1 ) n ψ n + 1 ( a ) ∑ k = 0 n ( − 1 ) k ( n k ) ( ln ( k + 1 ) ) m k + 1 , ℜ ( a ) > − 1 {\displaystyle \gamma _{m}=-{\frac {(\ln(1+a))^{m+1}}{m+1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {(\ln(k+1))^{m}}{k+1}},\quad \Re (a)>-1} and
γ m = − 1 r ( m + 1 ) ∑ l = 0 r − 1 ( ln ( 1 + a + l ) ) m + 1 + 1 r ∑ n = 0 ∞ ( − 1 ) n N n + 1 , r ( a ) ∑ k = 0 n ( − 1 ) k ( n k ) ( ln ( k + 1 ) ) m k + 1 , ℜ ( a ) > − 1 , r = 1 , 2 , 3 , … {\displaystyle \gamma _{m}=-{\frac {1}{r(m+1)}}\sum _{l=0}^{r-1}(\ln(1+a+l))^{m+1}+{\frac {1}{r}}\sum _{n=0}^{\infty }(-1)^{n}N_{n+1,r}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {(\ln(k+1))^{m}}{k+1}},\quad \Re (a)>-1,\;r=1,2,3,\ldots } or
γ m = − 1 1 2 + a { ( − 1 ) m m + 1 ζ ( m + 1 ) ( 0 , 1 + a ) − ( − 1 ) m ζ ( m ) ( 0 ) − ∑ n = 0 ∞ ( − 1 ) n ψ n + 2 ( a ) ∑ k = 0 n ( − 1 ) k ( n k ) ( ln ( k + 1 ) ) m k + 1 } , ℜ ( a ) > − 1 {\displaystyle \gamma _{m}=-{\frac {1}{{\tfrac {1}{2}}+a}}\left\{{\frac {(-1)^{m}}{m+1}}\,\zeta ^{(m+1)}(0,1+a)-(-1)^{m}\zeta ^{(m)}(0)-\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {(\ln(k+1))^{m}}{k+1}}\right\},\quad \Re (a)>-1} where ψn (a )Bernoulli polynomials of the second kind and Nn,r (a )
( 1 + z ) a + m − ( 1 + z ) a ln ( 1 + z ) = ∑ n = 0 ∞ N n , m ( a ) z n , | z | < 1 , {\displaystyle {\frac {(1+z)^{a+m}-(1+z)^{a}}{\ln(1+z)}}=\sum _{n=0}^{\infty }N_{n,m}(a)z^{n},\qquad |z|<1,} respectively (note that Nn,1 (a ) = ψn (a )
∑ 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 obtained slowly-convergent series involving unsigned Stirling numbers of the first kind [ ⋅ ⋅ ] {\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,...,} as well as semi-convergent series with rational terms only
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,} where m =0,1,2,... In particular, series for the first Stieltjes constant has a surprisingly simple form
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,} where H n n th harmonic number. More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, and Coffey.
Bounds and asymptotic growth The Stieltjes constants satisfy the bound
| 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}}} given by Berndt in 1972. Better bounds in terms of elementary functions were obtained by Lavrik
| 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 } by Israilov
| 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 } with k =1,2,... and C (1)=1/2, C (2)=7/12,... , by Nan-You and Williams
| 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}}} by Blagouchine
− | 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}}} where B n Bernoulli numbers , and by Matsuoka
| 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 } As concerns estimations resorting to non-elementary functions and solutions, Knessl, Coffey and Fekih-Ahmed obtained quite accurate results. For example, Knessl and Coffey give the following formula that approximates the Stieltjes constants relatively well for large n . If v is the unique solution of
2 π exp ( v tan v ) = n cos ( v ) v {\displaystyle 2\pi \exp(v\tan v)=n{\frac {\cos(v)}{v}}} with 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)} where
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).} Up to n = 100000, the Knessl-Coffey approximation correctly predicts the sign of γn
Namely, when n >> 1 {\displaystyle n>>1}
γ n ∼ 2 π n ! R e Γ ( s n ) e − c s n ( s n ) n n + s n + 3 2 {\displaystyle \gamma _{n}\sim {\sqrt {\frac {2}{\pi }}}n!\mathrm {Re} {\frac {\Gamma \left(s_{n}\right)e^{-cs_{n}}}{\left(s_{n}\right)^{n}{\sqrt {n+s_{n}+{\frac {3}{2}}}}}}} where s n {\displaystyle s_{n}}
s n = n + 3 2 W ( ± n + 3 2 2 π i ) {\displaystyle s_{n}={\frac {n+{\frac {3}{2}}}{W\left(\pm {\frac {n+{\frac {3}{2}}}{2\pi i}}\right)}}} W {\displaystyle W} c {\displaystyle c}
c = log ( 2 π ) + π 2 i {\displaystyle c=\log(2\pi )+{\frac {\pi }{2}}i} Defining a complex "phase" φ n {\displaystyle \varphi _{n}}
φ n ≡ 1 2 ln ( 8 π ) − n + ( n + 1 2 ) ln ( n ) + ( s n − n − 1 2 ) ln ( s n ) − 1 2 ln ( n + s n ) − ( c + 1 ) s n {\displaystyle \varphi _{n}\equiv {\frac {1}{2}}\ln(8\pi )-n+(n+{\frac {1}{2}})\ln(n)+(s_{n}-n-{\frac {1}{2}})\ln \left(s_{n}\right)-{\frac {1}{2}}\ln \left(n+s_{n}\right)-(c+1)s_{n}} we get a particularly simple expression in which both the rapidly increasing amplitude and the oscillations are clearly seen:
γ n ∼ R e [ e φ n ] = e R e φ n cos ( I m φ n ) {\displaystyle \gamma _{n}\sim \mathrm {Re} \left[e^{\varphi _{n}}\right]=e^{\mathrm {Re} \varphi _{n}}\cos \left(\mathrm {Im} \varphi _{n}\right)}
Numerical values The first few values are
n approximate value of γn 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
For large n , the Stieltjes constants grow rapidly in absolute value, and change signs in a complex pattern.
Further information related to the numerical evaluation of Stieltjes constants may be found in works of Keiper, Kreminski, Plouffe, Johansson and Blagouchine. First, Johansson provided values of the Stieltjes constants up to n = 100000, accurate to over 10000 digits each (the numerical values can be retrieved from the LMFDB [1]. Later, Johansson and Blagouchine devised a particularly efficient algorithm for computing generalized Stieltjes constants (see below) for large n a γ n n n =10100
Generalized Stieltjes constants
More generally, one can define Stieltjes constants γn Laurent series expansion of the Hurwitz zeta function :
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}.} Here a is a complex number with Re(a )>0. Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have γn n 0 (a)=-Ψ(a), while other constants are not known to be reducible to any elementary or classical function of analysis. Nevertheless, there are numerous representations for them. For example, there exists the following asymptotic representation
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}}} due to Berndt and Wilton. The analog of Jensen-Franel's formula for the generalized Stieltjes constant is the Hermite formula
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 , … ℜ ( a ) > 0 {\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]\Re (a)>0\end{array}}} Similar representations are given by the following formulas:
γ n ( a ) = − ( ln ( a − 1 2 ) ) n + 1 n + 1 + i ∫ 0 ∞ d x e 2 π x + 1 { ( ln ( a − 1 2 − i x ) ) n a − 1 2 − i x − ( ln ( a − 1 2 + i x ) ) n a − 1 2 + i x } , n = 0 , 1 , 2 , … ℜ ( a ) > 1 2 {\displaystyle \gamma _{n}(a)=-{\frac {{\big (}\ln(a-{\frac {1}{2}}){\big )}^{n+1}}{n+1}}+i\int _{0}^{\infty }{\frac {dx}{e^{2\pi x}+1}}\left\{{\frac {{\big (}\ln(a-{\frac {1}{2}}-ix){\big )}^{n}}{a-{\frac {1}{2}}-ix}}-{\frac {{\big (}\ln(a-{\frac {1}{2}}+ix){\big )}^{n}}{a-{\frac {1}{2}}+ix}}\right\},\qquad {\begin{array}{l}n=0,1,2,\ldots \\[1mm]\Re (a)>{\frac {1}{2}}\end{array}}} and
γ n ( a ) = − π 2 ( n + 1 ) ∫ 0 ∞ ( ln ( a − 1 2 − i x ) ) n + 1 + ( ln ( a − 1 2 + i x ) ) n + 1 ( cosh ( π x ) ) 2 d x , n = 0 , 1 , 2 , … ℜ ( a ) > 1 2 {\displaystyle \gamma _{n}(a)=-{\frac {\pi }{2(n+1)}}\int _{0}^{\infty }{\frac {{\big (}\ln(a-{\frac {1}{2}}-ix){\big )}^{n+1}+{\big (}\ln(a-{\frac {1}{2}}+ix){\big )}^{n+1}}{{\big (}\cosh(\pi x){\big )}^{2}}}\,dx,\qquad {\begin{array}{l}n=0,1,2,\ldots \\[1mm]\Re (a)>{\frac {1}{2}}\end{array}}} Generalized Stieltjes constants satisfy the following recurrence relation
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}}} as well as the multiplication theorem
∑ 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 } where ( p r ) {\displaystyle {\binom {p}{r}}} binomial coefficient (see and, pp. 101–102).
First generalized Stieltjes constant The first generalized Stieltjes constant has a number of remarkable properties.
Malmsten's identity (reflection formula for the first generalized Stieltjes constants): the reflection formula for the first generalized Stieltjes constant has the following form 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}}} where m and n are positive integers such that m <n . This formula has been long-time attributed to Almkvist and Meurman who derived it in 1990s. However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.
Rational arguments theorem: the first generalized Stieltjes constant at rational argument may be evaluated in a quasi-closed form via the following formula: 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\,.} see Blagouchine. An alternative proof was later proposed by Coffey and several other authors.
Finite summations: there are numerous summation formulae for the first generalized Stieltjes constants. For example, ∑ 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}}} For more details and further summation formulae, see.
Some particular values: some particular values of the first generalized Stieltjes constant at rational arguments may be reduced to the gamma-function, the first Stieltjes constant, and elementary functions. For instance, 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 } At the points 1/4, 3/4, and 1/3, values of first generalized Stieltjes constants were independently obtained by Connon and Blagouchine:
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}}} At points 2/3, 1/6, and 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}}} These values were calculated by Blagouchine, due to whom we also have the following:
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 3 ( 3 ln 2 + ln 3 + ln π ) ⋅ ln ( 1 + 3 ) − 7 − 6 3 2 ( ln 2 ) 2 − 3 4 ( ln 3 ) 2 + 3 3 ( 1 − 3 ) 2 ln 3 ⋅ ln 2 + 3 ln 2 ⋅ ln π − π ( 17 + 8 3 ) 2 3 ln 2 + π ( 1 − 3 ) 3 4 ln 3 − π 3 ( 2 + 3 ) ln π = − 29.84287823 … {\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}}}
Second generalized Stieltjes constant The second generalized Stieltjes constant is much less studied than the first constant. Similarly to the first generalized Stieltjes constant, the second generalized Stieltjes constant at rational argument may be evaluated via the following formula:
γ 2 ( r m ) = γ 2 + 2 3 ∑ l = 1 m − 1 cos 2 π r l m ⋅ ζ ‴ ( 0 , l m ) − 2 ( γ + ln 2 π m ) ∑ l = 1 m − 1 cos 2 π r l m ⋅ ζ ″ ( 0 , l m ) + π ∑ l = 1 m − 1 sin 2 π r l m ⋅ ζ ″ ( 0 , l m ) − 2 π ( γ + ln 2 π m ) ∑ l = 1 m − 1 sin 2 π r l m ⋅ ln Γ ( l m ) − 2 γ 1 ln m − γ 3 − [ ( γ + ln 2 π m ) 2 − π 2 12 ] ⋅ Ψ ( r m ) + π 3 12 cot π r m − γ 2 ln ( 4 π 2 m 3 ) + π 2 12 ( γ + ln m ) − γ ( ( ln 2 π ) 2 + 4 ln m ⋅ ln 2 π + 2 ( ln m ) 2 ) − { ( ln 2 π ) 2 + 2 ln 2 π ⋅ ln m + 2 3 ( ln m ) 2 } ln m , r = 1 , 2 , 3 , … , m − 1. {\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}}\,,\qquad \quad r=1,2,3,\ldots ,m-1.} see Blagouchine. An equivalent result was later obtained by Coffey by another method.