In mathematics, the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the 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 } is known as the Euler–Mascheroni constant.
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 is the 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 Gn are Gregory's coefficients, also known as reciprocal logarithmic numbers (G1=+1/2, G2=−1/12, G3=+1/24, G4=−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) are the Bernoulli polynomials of the second kind and Nn,r(a) are the polynomials given by the generating equation
( 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)). Oloa and Tauraso showed that series with harmonic numbers may lead to Stieltjes constants
∑ n = 1 ∞ H n − ( γ + ln n ) n = − γ 1 − 1 2 γ 2 + 1 12 π 2 ∑ n = 1 ∞ H n