thumb|The area of the blue region converges on the [[Euler–Mascheroni constant, which is the 0th Stieltjes constant.]]
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.
Representations
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 0<sup>0</sup>, 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.
As concerns series representations, a famous series employing an integer part of a logarithm was given by Hardy in 1912
: <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 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, Blagouchine
: <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 are the Bernoulli polynomials of the second kind and 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 ).
Oloa and Tauraso 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 Better bounds in terms of elementary functions were obtained by Lavrik
:<math>
|\gamma_n| \leq \frac{n!}{2^{n+1,\qquad n=1, 2, 3,\ldots
</math>
by Israilov
:<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
:<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 B<sub>n</sub> are Bernoulli numbers, and by Matsuoka
:<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 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. 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 values
The first few values are
:{| class="wikitable"
| n || approximate value of γ<sub>n</sub> || 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<sup>17</sup> ||
|-
| 1000 || −1.5709538442047449345494023425120825242380299554 × 10<sup>486</sup> ||
|-
| 10000 || −2.2104970567221060862971082857536501900234397174 × 10<sup>6883</sup> ||
|-
| 100000 || +1.9919273063125410956582272431568589205211659777 × 10<sup>83432</sup> ||
|}
The n-th Stieltjes constant is positive for
: n = 0, 3, 4, 5, 10, 11, 12, 17, 18, 19, 20, 21, 26, 27, 28, 29, 30, ...
and is negative for
: n = 1, 2, 6, 7, 8, 9, 13, 14, 15, 16, 22, 23, 24, 25, 31, 32, 33, 34, 35, ...
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
:<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 and, 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:
:<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. 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,
:<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. and Blagouchine:
:<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,