In mathematics, the Stieltjes constants are the numbers <math>\gamma_k</math> that occur in the Laurent series expansion of the Riemann zeta function:
- <math>\zeta(1+s)=\frac{1}{s}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n s^n.</math>
The constant <math>\gamma_0 = \gamma = 0.577\dots</math> is known as the Euler–Mascheroni constant.
RepresentationsEdit
The Stieltjes constants are given by the limit
- <math> \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\}}. </math>
(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
- <math>\gamma_n = \frac{(-1)^n n!}{2\pi} \int_0^{2\pi} e^{-nix} \zeta\left(e^{ix}+1\right) dx.</math>
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.<ref name="cpo">Template:Cite journal</ref><ref name="cf1">Template:Cite arXiv</ref><ref name="cf2">Template:Cite journal</ref><ref>Template:Cite journal</ref><ref name="blg_02"/><ref name="blg_03"/> In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that
- <math>
\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 </math> where δn,k is the Kronecker symbol (Kronecker delta).<ref name="blg_02"/><ref name="blg_03"/> Among other formulae, we find
- <math>
\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 </math>
- <math>
\begin{array}{l} \displaystyle \gamma_1 =-\left[\gamma -\frac{\ln2}{2}\right]\ln2 + 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} </math> see.<ref name="cpo"/><ref name="blg_02"/><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912<ref>Template:Cite journal</ref>
- <math>
\gamma_1 = \frac{\ln2}{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) </math> Israilov<ref name="isrl">Template:Cite journal</ref> gave semi-convergent series in terms of Bernoulli numbers <math>B_{2k}</math>
- <math>
\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]^{(2k-1)}_{x=n} - \theta\cdot\frac{B_{2N}}{(2N)!}\left[\frac{(\ln x)^m}{x}\right]^{(2N-1)}_{x=n} \,,\qquad 0<\theta<1 </math> Connon,<ref>Donal F. Connon Some applications of the Stieltjes constants, arXiv:0901.2083</ref> Blagouchine<ref name="blg_03"/><ref name="blg_04" /> and Coppo<ref name="cpo"/> gave several series with the binomial coefficients
- <math>
\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} </math> 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<ref name="blg_04">Template:Cite journal</ref>
- <math>
\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 </math> and
- <math>
\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 </math> or
- <math>
\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 </math> where Template:Math are the Bernoulli polynomials of the second kind and Template:Math are the polynomials given by the generating equation
- <math>
\frac{(1+z)^{a+m}-(1+z)^{a}}{\ln(1+z)}=\sum_{n=0}^\infty N_{n,m}(a) z^n , \qquad |z|<1, </math> respectively (note that Template:Math).<ref>Actually Blagouchine gives more general formulas, which are valid for the generalized Stieltjes constants as well.</ref> Oloa and Tauraso<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> showed that series with harmonic numbers may lead to Stieltjes constants
- <math>
\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^2_n - (\gamma+\ln n)^2}{n} = -\gamma_2 -2\gamma\gamma_1 -\frac{2}{3}\gamma^3+\frac{5}{3}\zeta(3) \end{array} </math> Blagouchine<ref name="blg_03" /> obtained slowly-convergent series involving unsigned Stirling numbers of the first kind <math>\left[{\cdot \atop \cdot}\right]</math>
- <math>
\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,..., </math> as well as semi-convergent series with rational terms only
- <math>
\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, </math> where m=0,1,2,... In particular, series for the first Stieltjes constant has a surprisingly simple form
- <math>
\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, </math> where Hn is the nth harmonic number.<ref name="blg_03" /> More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey.<ref name="cf1"/><ref name="cf2"/><ref name="blg_03" />
Bounds and asymptotic growthEdit
The Stieltjes constants satisfy the bound
- <math>
|\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} </math> given by Berndt in 1972.<ref>Bruce C. Berndt. On the Hurwitz Zeta-function. Rocky Mountain Journal of Mathematics, vol. 2, no. 1, pp. 151-157, 1972.</ref> Better bounds in terms of elementary functions were obtained by Lavrik<ref>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.</ref>
- <math>
|\gamma_n| \leq \frac{n!}{2^{n+1}},\qquad n=1, 2, 3,\ldots </math> by Israilov<ref name="isrl"/>
- <math>
|\gamma_n| \leq \frac{n! C(k)}{(2k)^{n}},\qquad n=1, 2, 3,\ldots </math> with k=1,2,... and C(1)=1/2, C(2)=7/12,... , by Nan-You and Williams<ref>Z. Nan-You and K. S. Williams. Some results on the generalized Stieltjes constants. Analysis, vol. 14, pp. 147-162, 1994.</ref>
- <math>
|\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} </math> by Blagouchine<ref name="blg_03">Template:Cite journal Corrigendum: vol. 173, pp. 631-632, 2017.</ref>
- <math>
\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} </math> where Bn are Bernoulli numbers, and by Matsuoka<ref>Y. Matsuoka. Generalized Euler constants associated with the Riemann zeta function. Number Theory and Combinatorics: Japan 1984, World Scientific, Singapore, pp. 279-295, 1985</ref><ref>Y. Matsuoka. On the power series coefficients of the Riemann zeta function. Tokyo Journal of Mathematics, vol. 12, no. 1, pp. 49-58, 1989.</ref>
- <math>
|\gamma_n| < 10^{-4} e^{n \ln \ln n}\,,\qquad n=5,6,7,\ldots </math>
As concerns estimations resorting to non-elementary functions and solutions, Knessl, Coffey<ref name="kns">Charles Knessl and Mark W. Coffey. An effective asymptotic formula for the Stieltjes constants. Math. Comp., vol. 80, no. 273, pp. 379-386, 2011.</ref> and Fekih-Ahmed<ref name="ahm">Lazhar Fekih-Ahmed. A New Effective Asymptotic Formula for the Stieltjes Constants, arXiv:1407.5567</ref> obtained quite accurate results. For example, Knessl and Coffey give the following formula that approximates the Stieltjes constants relatively well for large n.<ref name="kns" /> If v is the unique solution of
- <math>2 \pi \exp(v \tan v) = n \frac{\cos(v)}{v}</math>
with <math>0 < v < \pi/2</math>, and if <math>u = v \tan v</math>, then
- <math>\gamma_n \sim \frac{B}{\sqrt{n}} e^{nA} \cos(an+b)</math>
where
- <math>A = \frac{1}{2} \ln(u^2+v^2) - \frac{u}{u^2+v^2}</math>
- <math>B = \frac{2 \sqrt{2\pi} \sqrt{u^2+v^2}}{[(u+1)^2+v^2]^{1/4}}</math>
- <math>a = \tan^{-1}\left(\frac{v}{u}\right) + \frac{v}{u^2+v^2}</math>
- <math>b = \tan^{-1}\left(\frac{v}{u}\right) - \frac{1}{2} \left(\frac{v}{u+1}\right).</math>
Up to n = 100000, the Knessl-Coffey approximation correctly predicts the sign of γn with the single exception of n = 137.<ref name="kns" />
In 2022 K. Maślanka<ref name="msl">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</ref> 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 <math>n >> 1</math>
- <math>\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}}}</math>
where <math>s_{n}</math> are the saddle points:
- <math>s_{n}=\frac{n+\frac{3}{2}}{W\left( \pm \frac{n+\frac{3}{2}}{2\pi i}\right) }</math>
<math>W</math> is the Lambert function and <math>c</math> is a constant:
- <math>c=\log (2\pi )+\frac{\pi }{2}i</math>
Defining a complex "phase" <math>\varphi_{n}</math>
- <math>\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}</math>
we get a particularly simple expression in which both the rapidly increasing amplitude and the oscillations are clearly seen:
- <math>\gamma _{n}\sim \mathrm{Re} \left[ e^{\varphi _{n}}\right] =e^{\mathrm{Re}\varphi_{n}}\cos \left(\mathrm{Im}\varphi _{n}\right)</math>
Numerical valuesEdit
The first few values are <ref>Template:Cite journal</ref>
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,<ref>Template:Cite journal</ref> Kreminski,<ref>Template:Cite journal</ref> Plouffe,<ref>Simon Plouffe. Stieltjes Constants, from 0 to 78, 256 digits each</ref> Johansson<ref>Template:Cite journal</ref><ref name="johansson1" /> and Blagouchine.<ref name="johansson1" /> 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 Template:Math and complex Template:Math, which can be also used for ordinary Stieltjes constants.<ref name="johansson1" /> In particular, it allows one to compute Template:Math to 1000 digits in a minute for any Template:Math up to Template:Math.
Generalized Stieltjes constantsEdit
General informationEdit
More generally, one can define Stieltjes constants γn(a) that occur in the Laurent series expansion of the Hurwitz zeta function:
- <math>\zeta(s,a)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(a) (s-1)^n.</math>
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(1)=γn The zeroth constant is simply the digamma-function γ0(a)=-Ψ(a),<ref name="mse"/> 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
- <math>
\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\neq0, -1, -2, \ldots \end{array} </math> due to Berndt and Wilton. The analog of Jensen-Franel's formula for the generalized Stieltjes constant is the Hermite formula<ref name="blg_02"/>
- <math>
\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} </math> Similar representations are given by the following formulas:<ref name="johansson1">Template:Cite journal</ref>
- <math>
\gamma_n(a) = - \frac{\big(\ln(a-\frac12)\big)^{n+1}}{n+1} +i\int_0^\infty \frac{dx}{e^{2\pi x}+1} \left\{ \frac{\big(\ln(a-\frac12-ix)\big)^n}{a-\frac12-ix} - \frac{\big(\ln(a-\frac12+ix)\big)^n}{a-\frac12+ix} \right\} , \qquad \begin{array}{l} n=0, 1, 2,\ldots \\[1mm] \Re(a)>\frac12 \end{array} </math> and
- <math>
\gamma_n(a) = -\frac{\pi}{2(n+1)}\int_0^\infty \frac{\big(\ln(a-\frac12-ix)\big)^{n+1} + \big(\ln(a-\frac12+ix)\big)^{n+1}}{\big(\cosh(\pi x)\big)^2} \, dx , \qquad \begin{array}{l} n=0, 1, 2,\ldots \\[1mm] \Re(a)>\frac12 \end{array} </math>
Generalized Stieltjes constants satisfy the following recurrence relation
- <math>
\gamma_n(a+1) = \gamma_n(a) - \frac{(\ln a)^n}{a} \,, \qquad \begin{array}{l} n=0, 1, 2,\ldots \\[1mm] a\neq0, -1, -2, \ldots \end{array} </math> as well as the multiplication theorem
- <math>
\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 </math> where <math>\binom{p}{r}</math> denotes the binomial coefficient (see<ref>Template:Cite arXiv</ref> and,<ref name="blg_01"/> pp. 101–102).
First generalized Stieltjes constantEdit
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
- <math>
\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 m l}{n} \cdot\ln\Gamma \biggl(\frac{l}{n} \biggr) -\pi(\gamma+\ln2\pi n)\cot\frac{m\pi}{n} </math> 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.<ref name="adm">V. Adamchik. A class of logarithmic integrals. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.</ref> However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.<ref name="blg_02">Template:Cite journal And vol. 151, pp. 276-277, 2015. Template:ArXiv</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
- Rational arguments theorem: the first generalized Stieltjes constant at rational argument may be evaluated in a quasi-closed form via the following formula
- <math>
\begin{array}{ll} \displaystyle \gamma_1 \biggl(\frac{r}{m} \biggr)
=& \displaystyle
\gamma_1 +\gamma^2 + \gamma\ln2\pi m + \ln2\pi\cdot\ln{m}+\frac{1}{2}(\ln m)^2 + (\gamma+\ln2\pi m)\cdot\Psi\left(\frac{r}{m}\right) \\[5mm] \displaystyle & \displaystyle\qquad +\pi\sum_{l=1}^{m-1} \sin\frac{2\pi r l}{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\,. </math> see Blagouchine.<ref name="blg_02" /><ref name="mse">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> An alternative proof was later proposed by Coffey<ref name="coff">Mark W. Coffey Functional equations for the Stieltjes constants, Template:ArXiv</ref> and several other authors.
- Finite summations: there are numerous summation formulae for the first generalized Stieltjes constants. For example,
- <math>
\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+\ln2+2\ln m)\ln2\\[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\ln2+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+\ln2\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+\ln2\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)\ln2\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+\ln2\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} </math> For more details and further summation formulae, see.<ref name="blg_02" /><ref name="blg_01" />
- 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,
- <math>
\gamma_1\left(\frac{1}{2}\right) = - 2\gamma\ln 2 - (\ln 2)^2 + \gamma_1 = -1.353459680\ldots </math> At points 1/4, 3/4 and 1/3, values of first generalized Stieltjes constants were independently obtained by Connon<ref>Donal F. Connon The difference between two Stieltjes constants, arXiv:0906.0277</ref> and Blagouchine<ref name="blg_01">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</ref>
- <math>
\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)\ln2 - \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)\ln2 + \frac{\gamma\pi}{2}+\gamma_1 = -0.3912989024\ldots \\[6mm] \displaystyle \gamma_1\left(\frac{1}{3} \right) = -\frac{3\gamma}{2}\ln3 - \frac{3}{4}(\ln 3)^2 + \frac{\pi}{4\sqrt{3}}\left\{\ln3 - 8\ln2\pi -2\gamma +12 \ln\Gamma\left(\frac{1}{3} \right) \right\} + \gamma_1 = -3.259557515\ldots \end{array} </math> At points 2/3, 1/6 and 5/6
- <math>
\begin{array}{l} \displaystyle \gamma_1\left(\frac{2}{3} \right) = -\frac{3\gamma}{2}\ln3 - \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}\ln3 - \frac{3}{4}(\ln 3)^2 - (\ln 2)^2 - (3\ln3+2\gamma)\ln2 + \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\ln3 + 11\ln2 + \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\ln3+2\gamma)\ln2 - \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\ln3 + 11\ln2 + \frac{15}{2}\ln\pi + 3\gamma \right\}+ \gamma_1 = -0.2461690038\ldots \end{array} </math> These values were calculated by Blagouchine.<ref name="blg_01" /> To the same author are also due
- <math>
\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\sqrt5}}{2} \ln\Gamma \biggl(\frac{1}{5} \biggr) \\[5mm] & \displaystyle + \frac{\pi\sqrt{10-2\sqrt5}}{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}\ln5 -\frac{\pi\sqrt{25+10\sqrt5}}{10} \right\}\cdot\gamma \\[5mm] & \displaystyle - \frac{\sqrt{5}}{2}\left\{\ln2+\ln5+\ln\pi+\frac{\pi\sqrt{25-10\sqrt5}}{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}\ln2\cdot\ln5 + \frac{\sqrt{5}}{2}\ln2\cdot\ln\pi+\frac{\sqrt{5}}{4}\ln5\cdot\ln\pi
- \frac{\pi\big(2\sqrt{25+10\sqrt5}+5\sqrt{25+2\sqrt5} \big)}{20}\ln2\\[5mm] & \displaystyle - \frac{\pi\big(4\sqrt{25+10\sqrt5}-5\sqrt{5+2\sqrt5} \big)}{40}\ln5 - \frac{\pi\big(5\sqrt{5+2\sqrt5}+\sqrt{25+10\sqrt5} \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-\sqrt2\big)\ln\Gamma \biggl(\frac{1}{4} \biggr) \\[5mm] & \displaystyle -\left\{\frac{1+\sqrt2}{2}\pi+4\ln{2} +\sqrt{2}\ln\big(1+\sqrt{2}\big) \right\}\cdot\gamma - \frac{1}{\sqrt{2}}\big(\pi+8\ln2+2\ln\pi\big)\cdot\ln\big(1+\sqrt{2}) \\[5mm] & \displaystyle
- \frac{7\big(4-\sqrt2\big)}{4}(\ln 2)^2 + \frac{1}{\sqrt{2}}\ln2\cdot\ln\pi
-\frac{\pi\big(10+11\sqrt2\big)}{4}\ln2
-\frac{\pi\big(3+2\sqrt2\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+\sqrt3}{2}\pi+\frac{3}{2}\ln3 -\sqrt3(1-\sqrt3)\ln{2} +2\sqrt{3}\ln\big(1+\sqrt{3}\big) \right\}\cdot\gamma \\[5mm] & \displaystyle - 2\sqrt3\big(3\ln2+\ln3 +\ln\pi\big)\cdot\ln\big(1+\sqrt{3})
- \frac{7-6\sqrt3}{2}(\ln 2)^2 - \frac{3}{4}(\ln 3)^2 \\[5mm]
& \displaystyle + \frac{3\sqrt3(1-\sqrt3)}{2}\ln3\cdot\ln2
+ \sqrt3\ln2\cdot\ln\pi
-\frac{\pi\big(17+8\sqrt3\big)}{2\sqrt3}\ln2 \\[5mm]
& \displaystyle
+\frac{\pi\big(1-\sqrt3\big)\sqrt3}{4}\ln3
-\pi\sqrt3(2+\sqrt3)\ln\pi
= -29.84287823\ldots \end{array} </math>
Second generalized Stieltjes constantEdit
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
- <math>
\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 r l}{m} \cdot\zeta\left(0,\frac{l}{m}\right) - 2 (\gamma+\ln2\pi m) \sum_{l=1}^{m-1} \cos\frac{2\pi r l}{m} \cdot\zeta\left(0,\frac{l}{m}\right) \\[6mm] \displaystyle \quad + \pi\sum_{l=1}^{m-1} \sin\frac{2\pi r l}{m} \cdot\zeta\left(0,\frac{l}{m}\right) -2\pi(\gamma+\ln2\pi m) \sum_{l=1}^{m-1} \sin\frac{2\pi r l}{m} \cdot\ln\Gamma \biggl(\frac{l}{m} \biggr)
- 2\gamma_1 \ln{m} \\[6mm]
\displaystyle\quad - \gamma^3 -\left[(\gamma+\ln2\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. </math> see Blagouchine.<ref name="blg_02" /> An equivalent result was later obtained by Coffey by another method.<ref name="coff" />