Stieltjes constants

In mathematics, the Stieltjes constants are the numbers $\gamma _{k}$ that occur in the Laurent series expansion of the Riemann zeta function:

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

The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant.

The constant $\gamma _{0}=\gamma =0.577\dots$ is known as the Euler–Mascheroni constant.

Representations

The Stieltjes constants are given by the limit

\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

\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.[1][2][3][4][5][6] In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that

\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 δ_n,k is the Kronecker symbol (Kronecker delta).[5][6] Among other formulae, we find

\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

{\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.[1][5][7]

As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912[8]

\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[9] gave semi-convergent series in terms of Bernoulli numbers $B_{2k}$

\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

Connon,[10] Blagouchine[6][11] and Coppo[1] gave several series with the binomial coefficients

{\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 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[11]

\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

\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

\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 $ψ n (a)$ are the Bernoulli polynomials of the second kind and $N n,r (a)$ are the polynomials given by the generating equation

{\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 $N n,1 (a) = ψ n (a)$ ).[12] Oloa and Tauraso[13] showed that series with harmonic numbers may lead to Stieltjes constants

{\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] obtained slowly-convergent series involving unsigned Stirling numbers of the first kind $\left[{\cdot \atop \cdot }\right]$

\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,...,

as well as semi-convergent series with rational terms only

\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

\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 is the nth harmonic number.[6] More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey.[2][3][6]

Bounds and asymptotic growth

The Stieltjes constants satisfy the bound

|\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.[14] Better bounds in terms of elementary functions were obtained by Lavrik[15]

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

by Israilov[9]

|\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[16]

|\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[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}}

where B_n are Bernoulli numbers, and by Matsuoka[17][18]

|\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[19] and Fekih-Ahmed[20] obtained quite accurate results. For example, Knessl and Coffey give the following formula that approximates the Stieltjes constants relatively well for large n.[19] If v is the unique solution of

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

with $0<v<\pi /2$ , and if $u=v\tan v$ , then

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

where

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}}}

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

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 γ_n with the single exception of n = 137.[19]

In 2022 K. Maślanka[21] gave an asymptotic expression for the Stieltjes constants, which is both simpler and more accurate than those previously known. In particular, it reproduces with a relatively small error the troublesome value for n = 137.

Namely, when $n>>1$

\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 $s_{n}$ are the saddle points:

s_{n}={\frac {n+{\frac {3}{2}}}{W\left(\pm {\frac {n+{\frac {3}{2}}}{2\pi i}}\right)}}

$W$ is the Lambert function and $c$ is a constant:

c=\log(2\pi )+{\frac {\pi }{2}}i

Defining a complex "phase" $\varphi _{n}$

\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:

\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 [22]

n	approximate value of γ_n	OEIS
0	+0.5772156649015328606065120900824024310421593359	A001620
1	−0.0728158454836767248605863758749013191377363383	A082633
2	−0.0096903631928723184845303860352125293590658061	A086279
3	+0.0020538344203033458661600465427533842857158044	A086280
4	+0.0023253700654673000574681701775260680009044694	A086281
5	+0.0007933238173010627017533348774444448307315394	A086282
6	−0.0002387693454301996098724218419080042777837151	A183141
7	−0.0005272895670577510460740975054788582819962534	A183167
8	−0.0003521233538030395096020521650012087417291805	A183206
9	−0.0000343947744180880481779146237982273906207895	A184853
10	+0.0002053328149090647946837222892370653029598537	A184854
100	−4.2534015717080269623144385197278358247028931053 × 10¹⁷
1000	−1.5709538442047449345494023425120825242380299554 × 10⁴⁸⁶
10000	−2.2104970567221060862971082857536501900234397174 × 10⁶⁸⁸³
100000	+1.9919273063125410956582272431568589205211659777 × 10⁸³⁴³²

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,[23] Kreminski,[24] Plouffe,[25] Johansson[26][27] and Blagouchine.[27] 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 . Later, Johansson and Blagouchine devised a particularly efficient algorithm for computing generalized Stieltjes constants (see below) for large $n$ and complex $a$ , which can be also used for ordinary Stieltjes constants.[27] In particular, it allows one to compute $γ n$ to 1000 digits in a minute for any $n$ up to $n =10 100$ .

Generalized Stieltjes constants

General information

More generally, one can define Stieltjes constants γ_n(a) that occur in the Laurent series expansion of the Hurwitz zeta function:

\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 γ_n(1)=γ_n The zeroth constant is simply the digamma-function γ₀(a)=-Ψ(a),[28] 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

\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[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]\Re (a)>0\end{array}}

Similar representations are given by the following formulas:[27]

\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

\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

\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

\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 ${\binom {p}{r}}$ denotes the binomial coefficient (see[29] and,[30] 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

\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.[31] However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.[5][32]

Rational arguments theorem: the first generalized Stieltjes constant at rational argument may be evaluated in a quasi-closed form via the following formula

{\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.[5][28] An alternative proof was later proposed by Coffey[33] and several other authors.

Finite summations: there are numerous summation formulae for the first generalized Stieltjes constants. For example,

{\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.[5][30]

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,

\gamma _{1}\left({\frac {1}{2}}\right)=-2\gamma \ln 2-(\ln 2)^{2}+\gamma _{1}=-1.353459680\ldots

At points 1/4, 3/4 and 1/3, values of first generalized Stieltjes constants were independently obtained by Connon[34] and Blagouchine[30]

{\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

{\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.[30] To the same author are also due

{\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

{\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.[5] An equivalent result was later obtained by Coffey by another method.[33]

References

Coppo, Marc-Antoine (1999). "Nouvelles expressions des constantes de Stieltjes". Expositiones Mathematicae. 17: 349–358.
Coffey, Mark W. (2009). "Series representations for the Stieltjes constants". arXiv:0905.1111 [math-ph].
Coffey, Mark W. (2010). "Addison-type series representation for the Stieltjes constants". J. Number Theory. 130 (9): 2049–2064. doi:10.1016/j.jnt.2010.01.003.
Choi, Junesang (2013). "Certain integral representations of Stieltjes constants". Journal of Inequalities and Applications. 532: 1–10.
Blagouchine, Iaroslav V. (2015). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009. And vol. 151, pp. 276-277, 2015. arXiv:1401.3724
Blagouchine, Iaroslav V. (2016). "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. 158: 365–396. arXiv:1501.00740. doi:10.1016/j.jnt.2015.06.012. Corrigendum: vol. 173, pp. 631-632, 2017.
"A couple of definite integrals related to Stieltjes constants". Stack Exchange.
Hardy, G. H. (2012). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.
Israilov, M. I. (1981). "On the Laurent decomposition of Riemann's zeta function [in Russian]". Trudy Mat. Inst. Akad. Nauk. SSSR. 158: 98–103.
Donal F. Connon Some applications of the Stieltjes constants, arXiv:0901.2083
Blagouchine, Iaroslav V. (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF). INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A (#A3): 1–45. arXiv:1606.02044.
Actually Blagouchine gives more general formulas, which are valid for the generalized Stieltjes constants as well.
"A closed form for the series ..." Stack Exchange.
Bruce C. Berndt. On the Hurwitz Zeta-function. Rocky Mountain Journal of Mathematics, vol. 2, no. 1, pp. 151-157, 1972.
A. F. Lavrik. On the main term of the divisor's problem and the power series of the Riemann's zeta function in a neighbourhood of its pole (in Russian). Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 142, pp. 165-173, 1976.
Z. Nan-You and K. S. Williams. Some results on the generalized Stieltjes constants. Analysis, vol. 14, pp. 147-162, 1994.
Y. Matsuoka. Generalized Euler constants associated with the Riemann zeta function. Number Theory and Combinatorics: Japan 1984, World Scientific, Singapore, pp. 279-295, 1985
Y. Matsuoka. On the power series coefficients of the Riemann zeta function. Tokyo Journal of Mathematics, vol. 12, no. 1, pp. 49-58, 1989.
Charles Knessl and 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
Krzysztof Maślanka. Asymptotic Properties of Stieltjes Constants. Computational Methods in Science and Technology, vol. 28 (2022), p.123-131; https://arxiv.org/abs/2210.07244v1
Choudhury, B. K. (1995). "The Riemann zeta-function and its derivatives". Proc. R. Soc. A. 450 (1940): 477–499. Bibcode:1995RSPSA.450..477C. doi:10.1098/rspa.1995.0096. S2CID 124034712.
Keiper, J.B. (1992). "Power series expansions of Riemann ζ-function". Math. Comp. 58 (198): 765–773. Bibcode:1992MaCom..58..765K. doi:10.1090/S0025-5718-1992-1122072-5.
Kreminski, Rick (2003). "Newton-Cotes integration for approximating Stieltjes generalized Euler constants". Math. Comp. 72 (243): 1379–1397. Bibcode:2003MaCom..72.1379K. doi:10.1090/S0025-5718-02-01483-7.
Simon Plouffe. Stieltjes Constants, from 0 to 78, 256 digits each
Johansson, Fredrik (2015). "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives". Num. Alg. 69 (2): 253–570. arXiv:1309.2877. doi:10.1007/s11075-014-9893-1. S2CID 10344040.
Johansson, Fredrik; Blagouchine, Iaroslav (2019). "Computing Stieltjes constants using complex integration". Mathematics of Computation. 88 (318): 1829–1850. arXiv:1804.01679. doi:10.1090/mcom/3401. S2CID 4619883.
"Definite integral". Stack Exchange.
Connon, Donal F. (2009). "New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions". arXiv:0903.4539 [math.CA].
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.
"Evaluation of a particular integral". Stack Exchange.
Mark W. Coffey Functional equations for the Stieltjes constants, arXiv:1402.3746
Donal F. Connon The difference between two Stieltjes constants, arXiv:0906.0277

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[cpo-1] Coppo, Marc-Antoine (1999). "Nouvelles expressions des constantes de Stieltjes". Expositiones Mathematicae. 17: 349–358.

[cf1-2] Coffey, Mark W. (2009). "Series representations for the Stieltjes constants". arXiv:0905.1111 [math-ph].

[cf2-3] Coffey, Mark W. (2010). "Addison-type series representation for the Stieltjes constants". J. Number Theory. 130 (9): 2049–2064. doi:10.1016/j.jnt.2010.01.003.

[4] Choi, Junesang (2013). "Certain integral representations of Stieltjes constants". Journal of Inequalities and Applications. 532: 1–10.

[blg_02-5] Blagouchine, Iaroslav V. (2015). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009. And vol. 151, pp. 276-277, 2015. arXiv:1401.3724

[blg_03-6] Blagouchine, Iaroslav V. (2016). "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. 158: 365–396. arXiv:1501.00740. doi:10.1016/j.jnt.2015.06.012. Corrigendum: vol. 173, pp. 631-632, 2017.

[7] "A couple of definite integrals related to Stieltjes constants". Stack Exchange.

[8] Hardy, G. H. (2012). "Note on Dr. Vacca's series for γ". Q. J. Pure Appl. Math. 43: 215–216.

[isrl-9] Israilov, M. I. (1981). "On the Laurent decomposition of Riemann's zeta function [in Russian]". Trudy Mat. Inst. Akad. Nauk. SSSR. 158: 98–103.

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

[blg_04-11] Blagouchine, Iaroslav V. (2018). "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF). INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A (#A3): 1–45. arXiv:1606.02044.

[12] Actually Blagouchine gives more general formulas, which are valid for the generalized Stieltjes constants as well.

[13] "A closed form for the series ..." Stack Exchange.

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

[15] A. F. Lavrik. On the main term of the divisor's problem and the power series of the Riemann's zeta function in a neighbourhood of its pole (in Russian). Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 142, pp. 165-173, 1976.

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

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

[18] 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-19] Charles Knessl and Mark W. Coffey. An effective asymptotic formula for the Stieltjes constants. Math. Comp., vol. 80, no. 273, pp. 379-386, 2011.

[ahm-20] Lazhar Fekih-Ahmed. A New Effective Asymptotic Formula for the Stieltjes Constants, arXiv:1407.5567

[msl-21] Krzysztof Maślanka. Asymptotic Properties of Stieltjes Constants. Computational Methods in Science and Technology, vol. 28 (2022), p.123-131; https://arxiv.org/abs/2210.07244v1

[22] Choudhury, B. K. (1995). "The Riemann zeta-function and its derivatives". Proc. R. Soc. A. 450 (1940): 477–499. Bibcode:1995RSPSA.450..477C. doi:10.1098/rspa.1995.0096. S2CID 124034712.

[23] Keiper, J.B. (1992). "Power series expansions of Riemann ζ-function". Math. Comp. 58 (198): 765–773. Bibcode:1992MaCom..58..765K. doi:10.1090/S0025-5718-1992-1122072-5.

[24] Kreminski, Rick (2003). "Newton-Cotes integration for approximating Stieltjes generalized Euler constants". Math. Comp. 72 (243): 1379–1397. Bibcode:2003MaCom..72.1379K. doi:10.1090/S0025-5718-02-01483-7.

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

[26] Johansson, Fredrik (2015). "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives". Num. Alg. 69 (2): 253–570. arXiv:1309.2877. doi:10.1007/s11075-014-9893-1. S2CID 10344040.

[johansson1-27] Johansson, Fredrik; Blagouchine, Iaroslav (2019). "Computing Stieltjes constants using complex integration". Mathematics of Computation. 88 (318): 1829–1850. arXiv:1804.01679. doi:10.1090/mcom/3401. S2CID 4619883.

[mse-28] "Definite integral". Stack Exchange.

[29] Connon, Donal F. (2009). "New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions". arXiv:0903.4539 [math.CA].

[blg_01-30] 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-31] V. Adamchik. A class of logarithmic integrals. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.

[32] "Evaluation of a particular integral". Stack Exchange.

[coff-33] Mark W. Coffey Functional equations for the Stieltjes constants, arXiv:1402.3746

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