Stieltjes constants explained

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) = (1)/(s) + \sum ^{inf ty} _{n =0} (( -1 ) ^{n})/(n!) \gamma _{n} s ^{n.}

The constant

\gamma _{0} = \gamma = 0 . 577... is known as the Euler - Mascheroni constant.

Representations

The Stieltjes constants are given by the limit

\gamma _{n} = \lim _{m\toinfty} \left\{\sum ^{m} _{k =1} (( ln k) ^{n})/(k) - \int ^{m (( ln x) ^{n})/(x)} _{1} dx\right\} = \lim _{m} {\left\{\sum ^{m} _{k =1} (( ln k) ^{n})/(k) - (( 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} = (( -1 ) ^{n} n!)/(2 \pi) \int ^{2 \pi} _{0} 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]} In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that

\gamma _{n} = (1)/(2) \delta _{n,0} + (1)/(i) \int ^{inf ty} _{0} (dx)/(\left\{ e ^{2\pi} -1) (( ln ( 1- ix)) ^{n})/(1- ix) - (( ln ( 1+ ix)) ^{n})/(1+ ix) \right\}, n =0 , 1 , 2 ,\ldots where δ_n,k is the Kronecker symbol (Kronecker delta). Among other formulae, we find\gamma _{n} =- (\pi)/(2 (n +1 )) \int ^{inf ty (\left( ln \left( (1)/(\pm ix\right)\right) ^{n+1} 2))/(\cosh ^{2} \pi x)} _{-inf ty} dx n =0 , 1 , 2 ,\ldots \begin{array}{l} \displaystyle \gamma _{1} =- \left[\gamma - (ln2)/(2) \right] ln2 + i\int ^{inf ty} _{0} (dx)/(\left\{ e ^{\pi} +1) (ln ( 1- ix))/(1- ix) - (ln ( 1+ ix))/(1+ ix) \right\} \\[ 6 mm] \displaystyle \gamma _{1} = - \gamma ^{2} - \int ^{inf ty} _{0} \left[ (1)/(- 1- e ^{-x}) (1)/(x) \right] e ^{-x} ln x dx \end{array} see.^{[1] [5]}

As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912^[6]

\gamma _{1} = (ln2)/(2) \sum ^{inf ty} _{k =2} (( -1 ) ^{k})/(k) \lfloor log _{2{k}\rfloor ⋅ \left(2log} _{2{k}} - \lfloor log _{2{2k}\rfloor\right)} Israilov^[7] gave semi-convergent series in terms of Bernoulli numbers B _{2k} \gamma _{m} = \sum ^{n} _{k =1} (( ln k) ^{m})/(k) - (( ln n) ^{m+1})/(m +1) - (( ln n) ^{m})/(2 n) - \sum ^{N -1} _{k =1} (B _{2k})/(\left[ ( 2 k)!) (( ln x) ^{m})/(x) \right] ^{( 2 k -1 )} _{x = n} - \theta ⋅ (B _{2N})/(\left[ ( 2 N)!) (( ln x) ^{m})/(x) \right] ^{( 2 N -1 )} _{x = n} , 0 <\theta< 1 Connon,^[8] Blagouchine and Coppo^[1] gave several series with the binomial coefficients\begin{array}{l} \displaystyle \gamma _{m} =- (1)/(m +1) \sum ^{inf ty (1)/(n +1)} _{n =0} \sum ^{n} _{k =0} ( -1 ) ^{k} \binom{n}{k}( ln (k +1 )) ^{m+1} \\[ 7 mm] \displaystyle \gamma _{m} =- (1)/(m +1) \sum ^{inf ty (1)/(n +2)} _{n =0} \sum ^{n} _{k =0} ( -1 ) ^{k} \binom{n}{k} (( ln (k +1 )) ^{m+1})/(k +1) \\[ 7 mm] \displaystyle \gamma _{m =- (1)/(m +1)} \sum ^{inf ty} _{n =0} H _{n+1} \sum ^{n} _{k =0} ( -1 ) ^{k} \binom{n}{k}( ln (k +2 )) ^{m+1} \\[ 7 mm] \displaystyle \gamma _{m} = \sum ^{inf ty\left|G} _{n +1} \right| \sum ^{n ( -1 )} _{k =0} ^{k} \binom{n}{k} (( 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^[9] \gamma _{m =- (( ln ( 1+ a)) ^{m+1})/(m +1)} + \sum ^{inf ty} _{n =0} ( -1 ) ^{n} \psi _{n+1} (a) \sum ^{n} _{k =0} ( -1 ) ^{k} \binom{n}{k} (( ln (k +1 )) ^{m})/(k +1) , \Re(a)> -1 and \gamma _{m =- (1)/(r(m +1 ))} \sum ^{r -1} _{l =0} ( ln ( 1+ a + l)) ^{m+1} + (1)/(r) \sum ^{inf ty} _{n =0} ( -1 ) ^{n} N _{n+1,r} (a) \sum ^{n} _{k =0} ( -1 ) ^{k} \binom{n}{k} (( ln (k +1 )) ^{m})/(k +1) , \Re(a)> -1 , r =1 , 2 , 3 ,\ldots or\gamma _{m =- (1)/(\tfrac{ 1)} { 2 } + a} \left\{ (( -1 ) ^{m})/(m +1) \zeta ^{(m+1)} ( 0 , 1+ a) - ( -1 ) ^{m} \zeta ^{(m)} ( 0 ) - \sum ^{inf ty} _{n =0} ( -1 ) ^{n} \psi _{n+2} (a) \sum ^{n} _{k =0} ( -1 ) ^{k} \binom{n}{k} (( ln (k +1 )) ^{m})/(k +1) \right\} , \Re(a)> -1 where are the Bernoulli polynomials of the second kind and are the polynomials given by the generating equation(( 1+ z) ^{a+m} - ( 1+ z) ^{a})/(ln ( 1+ z)) = \sum ^{inf ty} _{n =0} N _{n,m} (a) z ^{n} , |z|< 1 , respectively (note that).^[10] Oloa and Tauraso^[11] showed that series with harmonic numbers may lead to Stieltjes constants\begin{array}{l} \displaystyle \sum ^{inf ty} _{n =1} (H _{n} - (\gamma +ln n))/(n) = - \gamma _{1} - (1)/(2) \gamma ^{2+ (1)/(12)} \pi ^{2} \\[ 6 mm] \displaystyle \sum ^{inf ty} _{n =1} (H ^{2} _{n} - (\gamma +ln n) ^{2})/(n) = - \gamma _{2} -2 \gamma\gamma _{1} - (2)/(3) \gamma ^{3+ (5)/(3)} \zeta( 3 ) \end{array} Blagouchine obtained slowly-convergent series involving unsigned Stirling numbers of the first kind\left[{ ⋅ \atop ⋅ }\right] \gamma _{m} = (1)/(2) \delta _{m,0} + (( -1 ) ^{m} m!)/(\pi) \sum ^{inf ty (1)/(n ⋅ n!)} _{n =1} \sum ^{\lfloor n/ 2 \rfloor} _{k =0} (( -1 ) ^{k} ⋅ \left[{ 2 k +2 \atop m +1)/(\right]) ⋅ \left[{n\atop 2 k +1 }\right]} {( 2 \pi) ^{2k+1} }\,,\qquad m=0,1,2,...,as well as semi-convergent series with rational terms only\gamma _{m} = (1)/(2) \delta _{m,0} + ( -1 ) ^{m} m! ⋅ \sum ^{N} _{k =1} (\left[{ 2 k\atop m +1)/(\right] ⋅) B _{2k} } + \theta\cdot\frac,\qquad 0<\theta<1,where m=0,1,2,... In particular, series for the first Stieltjes constant has a surprisingly simple form\gamma _{1} =- (1)/(2) \sum ^{N} _{k =1} (B _{2k} ⋅ H _{2k-1})/(k) + \theta ⋅ (B _{2N+2} ⋅ H _{2N+1})/(2 N +2) , 0 <\theta< 1 , where H_n is the nth harmonic number.More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey.^{[2] [3]}

Bounds and asymptotic growth

The Stieltjes constants satisfy the bound

|\gamma _{n|} \leq \begin{cases} \displaystyle (2 (n -1 )!)/(\pi ^{n}) , & n =1 , 3 , 5 ,\ldots \\[ 3 mm] \displaystyle (4 (n -1 )!)/(\pi ^{n}) , & n =2 , 4 , 6 ,\ldots \end{cases} given by Berndt in 1972.^[12] Better bounds in terms of elementary functions were obtained by Lavrik^[13] |\gamma _{n|} \leq (n!)/(2 ^{n+1}) , n =1 , 2 , 3 ,\ldots by Israilov^[7] |\gamma _{n|} \leq (n! C(k))/(( 2 k) ^{n}) , n =1 , 2 , 3 ,\ldots with k=1,2,... and C(1)=1/2, C(2)=7/12,..., by Nan-You and Williams^[14] |\gamma _{n|} \leq \begin{cases} \displaystyle (2 ( 2 n)!)/(n ^{n+1} ( 2 \pi) ^{n}) , & n =1 , 3 , 5 ,\ldots \\[ 4 mm] \displaystyle (4 ( 2 n)!)/(n ^{n+1} ( 2 \pi) ^{n}) , & n =2 , 4 , 6 ,\ldots \end{cases} by Blagouchine^[15] \begin{array}{ll} \displaystyle - (| {B)/(_{m+1}) | }{m +1 } < \gamma _{m} < (( 3 m +8 ) ⋅ | {B)/(_{m+3}) | }{ 24 } - (| {B)/(_{m+1}) | }{m +1 } , & m =1 , 5 , 9 ,\ldots\\[ 12 pt] \displaystyle (| B _{m+1} |)/(m +1) - (( 3 m +8 ) ⋅ | B _{m+3} |)/(24) < \gamma _{m} < (| {B)/(_{m+1}) | }{m +1 } , & m =3 , 7 , 11 ,\ldots\\[ 12 pt] \displaystyle - (| {B)/(_{m+2}) | }{ 2 } < \gamma _{m} < ((m +3 )(m +4 ) ⋅ | {B)/(_{m+4}) | }{ 48 } - (| B _{m+2} |)/(2) , & m =2 , 6 , 10 , \ldots\\[ 12 pt] \displaystyle (| {B)/(_{m+2}) | }{ 2 } - ((m +3 )(m +4 ) ⋅ | {B)/(_{m+4}) | }{ 48 } < \gamma _{m} < (| {B)/(_{m+2}) | }{ 2 }, & m =4 , 8 , 12 , \ldots\\ \end{array} where B_n are Bernoulli numbers, and by Matsuoka^{[16] [17]} |\gamma _{n|} < 10 ^{-4} e ^{n} , n =5 , 6 , 7 ,\ldots

As concerns estimations resorting to non-elementary functions and solutions, Knessl, Coffey^[18] and Fekih-Ahmed^[19] 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 \pi \exp(v \tan v) = n (\cos(v))/(v)

with

0 < v < \pi/ 2 , and if u = v \tan v , then\gamma _{n} \sim (B)/(\sqrt{n) } e^ \cos(an+b)

where

A = (1)/(2) ln (u ^{2+v} ^{2)} - (u)/(u ^{2+v} ^{2}) B = (2 \sqrt{ 2 \pi)/(\sqrt{u) ^{2+v} ^{2}}{[(u+1)} ^{2+v} ^{2]} ^{1/4} }a = \tan ^{-1} \left( (v)/(u) \right) + (v)/(u ^{2+v} ^{2}) b = \tan ^{-1} \left( (v)/(u) \right) - (1)/(\left( 2) (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.

In 2022 K. Maślanka^[20] 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 thetroublesome value for n = 137.

Namely, when

n >> 1 \gamma _{n} \sim \sqrt{ (2)/(\pi) } n! \mathrm \frac

where

s _{n} are the saddle points:s _{n} = (n + (3)/(2))/(W\left( \pm (n + (3)/(2))/(2 \pi i) \right)) W is the Lambert function and c is a constant:c =log ( 2 \pi ) + (\pi)/(2) i

Defining a complex "phase"

\varphi _{n} \varphi _{n} \equiv (1)/(2) ln ( 8 \pi ) - n + (n + (1)/(2) ) ln (n) + (s _{n} - n - (1)/(2) ) ln \left( s _{n} \right) - (1)/(2) ln \left( n + s _{n} \right) - (c +1 )s _{n}

we get a particularly simple expression in which both the rapidly increasingamplitude and the oscillations are clearly seen:

\gamma _{n} \sim Re \left[ e ^{\varphi _{n}} \right] = e ^{Re \varphi _{n}} \cos \left( Im \varphi _{n} \right)

Numerical values

The first few values are ^[21]

n	approximate value of γn	OEIS
0	+0.5772156649015328606065120900824024310421593359
1	-0.0728158454836767248605863758749013191377363383
2	-0.0096903631928723184845303860352125293590658061
3	+0.0020538344203033458661600465427533842857158044
4	+0.0023253700654673000574681701775260680009044694
5	+0.0007933238173010627017533348774444448307315394
6	-0.0002387693454301996098724218419080042777837151
7	-0.0005272895670577510460740975054788582819962534
8	-0.0003521233538030395096020521650012087417291805
9	-0.0000343947744180880481779146237982273906207895
10	+0.0002053328149090647946837222892370653029598537
100	-4.2534015717080269623144385197278358247028931053 × 10^17
1000	-1.5709538442047449345494023425120825242380299554 × 10^486
10000	-2.2104970567221060862971082857536501900234397174 × 10^6883
100000	+1.9919273063125410956582272431568589205211659777 × 10^83432

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,^[22] Kreminski,^[23] Plouffe,^[24] Johansson^[25] 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 http://beta.lmfdb.org/riemann/stieltjes/. Later, Johansson and Blagouchine devised a particularly efficient algorithm for computing generalized Stieltjes constants (see below) for large and complex, which can be also used for ordinary Stieltjes constants. In particular, it allows one to compute to 1000 digits in a minute for any up to .

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) = (1)/(s -1) + \sum ^{inf ty} _{n =0} (( -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), 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\toinfty} \left\{ \sum ^{m} _{k =0} (( ln (k + a)) ^{n})/(k + a) - (( ln (m + a)) ^{n+1})/(n +1) \right\}, \begin{array}{l} n =0 , 1 , 2 ,\ldots \\[ 1 mm] a ≠ 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\gamma _{n(a)} = \left[ (1)/(- 2 a) (ln {a)/() } \right](\ln a)^n-i\int_0^\infty \frac \left\, \qquad\beginn=0, 1, 2,\ldots \\[1mm]\Re(a)>0\endSimilar representations are given by the following formulas:^[26] \gamma _{n(a)} = - (( ln (a - (12 ) ))/(^{n+1}))/(n +1) + i\int ^{inf ty} _{0} (dx)/(\left\{ e ^{2\pi} +1) (( ln (a - (12- ix) ))/(^{n}))/(a - (12- ix)/()) - (( ln (a - (12+ ix) ))/(^{n}))/(a - (12+ ix)/()) \right\} , \begin{array}{l} n =0 , 1 , 2 ,\ldots \\[ 1 mm] \Re(a)> (12 \end{array})/() and\gamma _{n(a)} =- (\pi)/(2 (n +1 )) \int ^{inf ty} _{0} (( ln (a - (12- ix) ))/(+ ( ln (a - (12+ ix) ))/(^{n+1}) ^{n+1}))/(( \cosh(\pi x) ) ^{2}) dx , \begin{array}{l} n =0 , 1 , 2 ,\ldots \\[ 1 mm] \Re(a)> (12 \end{array})/()

Generalized Stieltjes constants satisfy the following recurrence relation

\gamma _{n(a+1)} = \gamma _{n(a)} - (( ln a) ^{n})/(a) , \begin{array}{l} n =0 , 1 , 2 ,\ldots \\[ 1 mm] a ≠ 0 , -1 , -2 , \ldots \end{array} as well as the multiplication theorem\sum ^{n -1} _{l =0} \gamma _{p} \left(a + (l)/(n) \right) = ( -1 ) ^{p} n \left[ (ln n)/(p +1) - \Psi(an) \right]( ln n) ^{p} + n\sum ^{p -1} _{r =0} ( -1 ) ^{r} \binom{p}{r} \gamma _{p-r} (an) ⋅ ( ln n) ^{r,} n =2 , 3 , 4 ,\ldots where \binom{p}{r} denotes the binomial coefficient (see^[27] 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

\gamma _{1} l ( (m)/(n) r ) - \gamma _{1} l ( 1- (m)/(n) r ) =2 \pi\sum ^{n -1} _{l =1} \sin (2 \pi m l)/(n) ⋅ ln \Gamma l ( (l)/(n) r ) - \pi(\gamma +ln2 \pi n)\cot (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.^[28] However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.^{[29] [30]}

• 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} l ( (r)/(m) r ) = & \displaystyle \gamma _{1} + \gamma ^{2} + \gamma ln2 \pi m +ln2 \pi ⋅ ln {m} + (1)/(2) ( ln m) ^{2 +} (\gamma +ln2 \pi m) ⋅ \Psi\left( (r)/(m) \right) \\[ 5 mm] \displaystyle & \displaystyle + \pi\sum ^{m -1} _{l =1} \sin (2 \pi r l)/(m) ⋅ ln \Gamma l ( (l)/(m) r ) + \sum ^{m -1} _{l =1} \cos (2 \pi rl)/(⋅ \zeta''\left( 0 , m) (l)/(m) \right) \end{array}, r =1 , 2 , 3 ,\ldots, m -1 . see Blagouchine.^[31] An alternative proof was later proposed by Coffey^[32] and several other authors.

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

\begin{array}{ll} \displaystyle \sum ^{m -1} _{r =0} \gamma _{1\left(} a + (r)/(m) \right) = m ln {m} ⋅ \Psi(am) - (m)/(2) ( ln m) ^{2} + m\gamma _{1(am),} a\in C \\[ 6 mm] \displaystyle \sum ^{m -1} _{r =1} \gamma _{1 \left( (r)/(m)} \right) = (m -1 )\gamma _{1} - m\gamma ln {m} - (m)/(2) ( ln m) ^{2} \\[ 6 mm] \displaystyle \sum ^{2 m -1} _{r =1} ( -1 ) ^{r} \gamma _{1} l ( (r)/(2 m) r ) = - \gamma _{1+m(2\gamma+ln2+2ln} m) ln2 \\[ 6 mm] \displaystyle \sum ^{2 m -1} _{r =0} ( -1 ) ^{r} \gamma _{1 l ( (2 r +1)/(4 m)} r ) = m\left\{ 4 \pi ln \Gamma l ( (1)/(4) r ) - \pi ( 4ln2+3ln \pi +ln m + \gamma ) \right\}\\[ 6 mm] \displaystyle \sum ^{m -1} _{r =1} \gamma _{1} l ( (r)/(m) r ) ⋅ \cos\dfrac{ 2 \pi rk}{m} = - \gamma _{1} + m(\gamma +ln2 \pi m) ln \left( 2 \sin (k\pi)/(m) \right) + (m)/(2) \left\{\zeta''\left( 0 , (k)/(m) \right) + \zeta''\left( 0 , 1- (k)/(m) \right) \right\}, k =1 , 2 ,\ldots,m -1 \\[ 6 mm] \displaystyle \sum ^{m -1} _{r =1} \gamma _{1 l ( (r)/(m)} r ) ⋅ \sin\dfrac{ 2 \pi rk}{m} = (\pi)/(2) (\gamma +ln2 \pi m)( 2 k - m) - (\pi m)/(2) \left\{ ln \pi -ln \sin (k\pi)/(m) \right\} + m\pi ln \Gamma l ( (k)/(m) r ) , k =1 , 2 ,\ldots,m -1 \\[ 6 mm] \displaystyle \sum ^{m -1} _{r =1} \gamma _{1} l ( (r)/(r ) ⋅ \cot m) (\pi r)/(m) = \displaystyle (\pi)/(6) \ {( 1- m)(m -2 )\gamma + 2 (m ^{2-1)ln2\pi} - (m ^{m -1} _{l =1} l ⋅ ln \Gamma\left( (l)/(m) \right) \\[ 6 mm] \displaystyle \sum ^{m -1} _{r =1} (r)/(m) ⋅ \gamma _{1} l ( (r)/(m) r ) = (1)/(2) \left\{(m -1 )\gamma _{1} - m\gamma ln {m} - (m)/(2) ( ln m) ^{2} \right\} - (\pi)/(2 m) (\gamma +ln2 \pi m) \sum ^{m -1} _{l =1} l ⋅ \cot (\pi l)/(- m) (\pi)/(2) \sum ^{m -1} _{l =1} \cot (\pi l)/(⋅ ln \Gamma l ( m) (l)/(m) r ) \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,

\gamma _{1 \left( (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^[33] and Blagouchine^[34] \begin{array}{l} \displaystyle \gamma _{1 \left( (1)/(4)} \right) =2 \pi ln \Gamma\left( (1)/(4) \right) - (3 \pi)/(2) ln \pi - (7)/(2) ( ln 2 ) ^{2} - ( 3 \gamma +2 \pi) ln2 - (\gamma\pi)/(2) + \gamma _{1} = -5 . 518076350 \ldots \\[ 6 mm] \displaystyle \gamma _{1 \left( (3)/(4)} \right) =-2 \pi ln \Gamma\left( (1)/(4) \right) + (3 \pi)/(2) ln \pi - (7)/(2) ( ln 2 ) ^{2} - ( 3 \gamma -2 \pi) ln2 + (\gamma\pi)/(2) + \gamma _{1} = -0 . 3912989024 \ldots \\[ 6 mm] \displaystyle \gamma _{1 \left( (1)/(3)} \right) =- (3 \gamma)/(2) ln3 - (3)/(4) ( ln 3 ) ^{2} + (\pi)/(4 \sqrt{ 3) }\left\+ \gamma_1 = -3.259557515\ldots\endAt points 2/3, 1/6 and 5/6\begin{array}{l} \displaystyle \gamma _{1 \left( (2)/(3)} \right) =- (3 \gamma)/(2) ln3 - (3)/(4) ( ln 3 ) ^{2} - (\pi)/(4 \sqrt{ 3) }\left\ + \gamma_1 = -0.5989062842\ldots \\[6mm]\displaystyle\gamma_1\left(\frac \right) = -\frac\ln3 - \frac(\ln 3)^2- (\ln 2)^2 - (3\ln3+2\gamma)\ln2 + \frac\ln\Gamma\left(\frac \right) \\[5mm]\displaystyle\qquad\qquad\quad- \frac\left\ + \gamma_1 = -10.74258252\ldots\\[6mm]\displaystyle\gamma_1\left(\frac \right) = -\frac\ln 3 - \frac(\ln 3)^2 - (\ln 2)^2 - (3\ln3+2\gamma)\ln2 - \frac\ln\Gamma\left(\frac \right) \\[6mm]\displaystyle\qquad\qquad\quad+ \frac\left\+ \gamma_1 = -0.2461690038\ldots\endThese values were calculated by Blagouchine. To the same author are also due \begin{array}{ll} \displaystyle \gamma _{1 l ( (1)/(5)} r ) = & \displaystyle \gamma _{1} + (\sqrt{ 5)/() }\left\+ \frac \ln\Gamma \biggl(\frac \biggr)\\[5mm]& \displaystyle + \frac \ln\Gamma \biggl(\frac \biggr)+\left\\cdot\gamma \\[5mm]& \displaystyle - \frac\left\\cdot\ln\big(1+\sqrt) +\frac(\ln 2)^2 + \frac(\ln 5)^2 \\[5mm]& \displaystyle +\frac\ln2\cdot\ln5 + \frac\ln2\cdot\ln\pi+\frac\ln5\cdot\ln\pi- \frac\ln2\\[5mm]& \displaystyle - \frac\ln5- \frac\ln\pi\\[5mm]& \displaystyle = -8.030205511\ldots \\[6mm]\displaystyle \gamma_1\biggl(\frac \biggr) =& \displaystyle\gamma_1 + \sqrt\left\+ 2\pi\sqrt\ln\Gamma \biggl(\frac \biggr)-\pi \sqrt\big(1-\sqrt2\big)\ln\Gamma \biggl(\frac \biggr)\\[5mm]& \displaystyle -\left\\cdot\gamma - \frac\big(\pi+8\ln2+2\ln\pi\big)\cdot\ln\big(1+\sqrt) \\[5mm]& \displaystyle - \frac(\ln 2)^2 + \frac\ln2\cdot\ln\pi -\frac\ln2 -\frac\ln\pi\\[5mm]& \displaystyle = -16.64171976\ldots \\[6mm]\displaystyle \gamma_1\biggl(\frac \biggr) =& \displaystyle\gamma_1 + \sqrt\left\+ 4\pi\ln\Gamma \biggl(\frac \biggr)+3\pi \sqrt\ln\Gamma \biggl(\frac \biggr)\\[5mm]& \displaystyle -\left\\cdot\gamma \\[5mm]& \displaystyle - 2\sqrt3\big(3\ln2+\ln3 +\ln\pi\big)\cdot\ln\big(1+\sqrt) - \frac(\ln 2)^2 - \frac(\ln 3)^2 \\[5mm]& \displaystyle + \frac\ln3\cdot\ln2 + \sqrt3\ln2\cdot\ln\pi -\frac\ln2 \\[5mm]& \displaystyle +\frac\ln3 -\pi\sqrt3(2+\sqrt3)\ln\pi= -29.84287823\ldots\end

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} l ( (r)/(m) r ) = \gamma _{2} + (2)/(3) \sum ^{m -1} _{l =1} \cos (2 \pi r l)/(⋅ \zeta'''\left( 0 , m) (l)/(m) \right) - 2 (\gamma +ln2 \pi m) \sum ^{m -1} _{l =1} \cos (2 \pi r l)/(⋅ \zeta''\left( 0 , m) (l)/(m) \right) \\[ 6 mm] \displaystyle + \pi\sum ^{m -1} _{l =1} \sin (2 \pi r l)/(⋅ \zeta''\left( 0 , m) (l)/(m) \right) -2 \pi(\gamma +ln2 \pi m) \sum ^{m -1} _{l =1} \sin (2 \pi r l)/(m) ⋅ ln \Gamma l ( (l)/(m) r ) - 2 \gamma _{1} ln {m} \\[ 6 mm] \displaystyle - \gamma ^{3} - \left[(\gamma +ln2 \pi m) ^{2- (\pi ^{2})/(12)} \right] ⋅ \Psi l ( (r)/(m) r ) + (\pi ^{3})/(\cot 12) (\pi r)/(m) - \gamma ^{2ln(4\pi} ^{2} m ^{3)} + (\pi ^{2})/(12) (\gamma +ln {m}) \\[ 6 mm] \displaystyle - \gamma ( ( ln 2 \pi) ^{2} + 4ln m ⋅ ln 2 \pi + 2 ( ln m) ^{2)} - \left\{( ln 2 \pi) ^{2} + 2ln 2 \pi ⋅ ln m + (2)/(3) ( ln m) ^{2\right\}ln} m \end{array}, r =1 , 2 , 3 ,\ldots, m -1 . see Blagouchine.An equivalent result was later obtained by Coffey by another method.

Notes and References

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