Stieltjes Constants

In mathematics, the Stieltjes constants are the numbers $\gamma_k$ that occur in the Laurent series expansion of the Riemann zeta function:

:$\zeta(1+s)=\frac+\sum_^\infty \frac \gamma_n s^n.$

The constant $\gamma_0 = \gamma = 0.577\dots$ is known as the Euler–Mascheroni constant.

Representations

The Stieltjes constants are given by the limit

: $\gamma_n = \lim_ \left\ = \lim_ .$

(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

:$\gamma_n = \frac \int_0^ e^ \zeta\left(e^+1\right) dx.$

Various representations in terms of integrals and infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors. In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that
: $\gamma_n = \frac\delta_+\frac\int_0^\infty \frac \left\\,, \qquad\quad n=0, 1, 2,\ldots$
where δ_''n,k'' is the Kronecker symbol (Kronecker delta). Among other formulae, we find
: $\gamma_n = -\frac \int_^\infty \frac\, dx \qquad\qquad\qquad\qquad\qquad\qquad n=0, 1, 2,\ldots$

:
\begin
\displaystyle
\gamma_1 =-\left gamma -\frac\rightln2 + i\int_0^\infty \frac \left\ \\ mm\displaystyle
\gamma_1 = -\gamma^2 - \int_0^\infty \left frac-\frac\righte^\ln x \, dx
\end

see.

As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912
: $\gamma_1 = \frac\sum_^\infty \frac \lfloor \log_2\rfloor\cdot \left(2\log_2 - \lfloor \log_2\rfloor\right)$
Israilov gave semi-convergent series in terms of Bernoulli numbers $B_$
:
\gamma_m = \sum_^n \frac - \frac
- \frac - \sum_^ \frac\left frac\right_
- \theta\cdot\frac\left frac\right_ \,,\qquad 0<\theta<1

Connon, Blagouchine and Coppo gave several series with the binomial coefficients
:
\begin
\displaystyle
\gamma_m = -\frac\sum_^\infty\frac
\sum_^n (-1)^k \binom(\ln(k+1))^ \\ mm\displaystyle
\gamma_m = -\frac\sum_^\infty\frac
\sum_^n (-1)^k \binom\frac \\ mm\displaystyle
\gamma_m=-\frac\sum_^\infty H_\sum_^n (-1)^k \binom(\ln(k+2))^\\ mm\displaystyle
\gamma_m = \sum_^\infty\left, G_\ \sum_^n
(-1)^k \binom\frac
\end

where ''G''_''n'' are Gregory's coefficients, also known a
reciprocal logarithmic numbers
(''G''₁=+1/2, ''G''₂=−1/12, ''G''₃=+1/24, ''G''₄=−19/720,... ).
More general series of the same nature include these examples
: $\gamma_m=-\frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom\frac,\quad \Re(a)>-1$
and
: $\gamma_m=-\frac\sum_^(\ln(1+a+l))^ + \frac\sum_^\infty (-1)^n N_(a) \sum_^ (-1)^k \binom\frac,\quad \Re(a)>-1, \; r=1,2,3,\ldots$
or
: $\gamma_m=-\frac \left\ ,\quad \Re(a)>-1$
where are the Bernoulli polynomials of the second kind and are the polynomials given by the generating equation
: $\frac=\sum_^\infty N_(a) z^n , \qquad , z, <1,$
respectively (note that ).
Oloa and Tauraso showed that series with harmonic numbers may lead to Stieltjes constants
:
\begin
\displaystyle
\sum_^\infty \frac =
-\gamma_1 -\frac\gamma^2+\frac\pi^2 \\ mm\displaystyle
\sum_^\infty \frac =
-\gamma_2 -2\gamma\gamma_1 -\frac\gamma^3+\frac\zeta(3)
\end

Blagouchine obtained slowly-convergent series involving unsigned Stirling numbers of the first kind
$\left$right/math>
: $\gamma_m = \frac\delta_+ \frac \sum_^\infty\frac \sum_^\frac \,,\qquad m=0,1,2,...,$
as well as semi-convergent series with rational terms only
: $\gamma_m = \frac\delta_+(-1)^ m!\cdot\sum_^\frac + \theta\cdot\frac,\qquad 0<\theta<1,$
where ''m''=0,1,2,... In particular, series for the first Stieltjes constant has a surprisingly simple form
: $\gamma_1 = -\frac\sum_^\frac + \theta\cdot\frac,\qquad 0<\theta<1,$
where ''H''_''n'' is the ''n''th harmonic number.
More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey.

Bounds and asymptotic growth

The Stieltjes constants satisfy the bound

:
, \gamma_n, \leq
\begin
\displaystyle \frac\,,\qquad & n=1, 3, 5,\ldots \\ mm\displaystyle \frac\,,\qquad & n=2, 4, 6,\ldots
\end

given by Berndt in 1972. Better bounds in terms of elementary functions were obtained by Lavrik
:$, \gamma_n, \leq \frac,\qquad n=1, 2, 3,\ldots$
by Israilov
:$, \gamma_n, \leq \frac,\qquad n=1, 2, 3,\ldots$
with ''k''=1,2,... and ''C''(1)=1/2, ''C''(2)=7/12,... , by Nan-You and Williams
:
, \gamma_n, \leq
\begin
\displaystyle \frac\,,\qquad & n=1, 3, 5,\ldots \\ mm\displaystyle \frac\,,\qquad & n=2, 4, 6,\ldots
\end

by Blagouchine Corrigendum: vol. 173, pp. 631-632, 2017.
:
\begin
\displaystyle-\frac < \gamma_m <
\frac - \frac ,
& m=1, 5, 9,\ldots\\ 2pt\displaystyle
\frac - \frac
< \gamma_m < \frac , & m=3, 7, 11,\ldots\\ 2pt\displaystyle -\frac < \gamma_m
< \frac - \frac ,
\qquad & m=2, 6, 10, \ldots\\ 2pt\displaystyle
\frac - \frac
< \gamma_m < \frac, & m=4, 8, 12, \ldots\\
\end

where ''B''_''n'' are Bernoulli numbers, and by Matsuoka
:$, \gamma_n, < 10^ e^\,,\qquad n=5,6,7,\ldots$

As concerns estimations resorting to non-elementary functions and solutions, Knessl, CoffeyCharles Knessl and Mark W. Coffey. ''An effective asymptotic formula for the Stieltjes constants''. Math. Comp., vol. 80, no. 273, pp. 379-386, 2011. and Fekih-AhmedLazhar 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''. If ''v'' is the unique solution of

:$2 \pi \exp(v \tan v) = n \frac$

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

:$\gamma_n \sim \frac e^ \cos(an+b)$

where

:$A = \frac \ln(u^2+v^2) - \frac$

:$B = \frac$

:$a = \tan^\left(\frac\right) + \frac$

:$b = \tan^\left(\frac\right) - \frac \left(\frac\right).$

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. MaślankaKrzysztof Maślanka. ''Asymptotic Properties of Stieltjes Constants''. Computational Methods in Science and Technology, vol. 28 (2022), p.123-131; https://arxiv.org/abs/2210.07244v1 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 $n >> 1$

:$\gamma_ \sim \sqrt n! \mathrm \frac$

where $s_$ are the saddle points:

:$s_=\frac$

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

:$c=\log (2\pi )+\fraci$

Defining a complex "phase" $\varphi_$

:$\varphi _\equiv \frac\ln (8\pi )-n+(n+\frac)\ln (n)+(s_-n-\frac)\ln \left( s_\right) -\frac\ln \left( n+s_\right)-(c+1)s_$

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

:\gamma _\sim \mathrm \left e^\right=e^\cos \left(\mathrm\varphi _\right)

Numerical values

The first few values are

:

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 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 LMFDBbr>
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:

:$\zeta(s,a)=\frac+\sum_^\infty \frac \gamma_n(a) (s-1)^n.$

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_\left\, \qquad
\begin
n=0, 1, 2,\ldots \\ mma\neq0, -1, -2, \ldots
\end

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 frac-\frac \right\ln a)^n
-i\int_0^\infty \frac \left\ , \qquad
\begin
n=0, 1, 2,\ldots \\ mm\Re(a)>0
\end

Similar representations are given by the following formulas:
:
\gamma_n(a) = - \frac
+i\int_0^\infty \frac \left\ , \qquad
\begin
n=0, 1, 2,\ldots \\ mm\Re(a)>\frac12
\end

and
:
\gamma_n(a) = -\frac\int_0^\infty \frac \, dx , \qquad
\begin
n=0, 1, 2,\ldots \\ mm\Re(a)>\frac12
\end

Generalized Stieltjes constants satisfy the following recurrence relation
:
\gamma_n(a+1) = \gamma_n(a) - \frac \,, \qquad
\begin
n=0, 1, 2,\ldots \\ mma\neq0, -1, -2, \ldots
\end

as well as the multiplication theorem
:
\sum_^ \gamma_p \left(a+\frac \right) =
(-1)^p n \left frac - \Psi(an) \right\ln n)^p + n\sum_^(-1)^r \binom \gamma_(an) \cdot (\ln n)^r\,,
\qquad\qquad n=2, 3, 4,\ldots

where $\binom$ denotes the binomial coefficient (see 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
:$\gamma_1 \biggl(\frac\biggr)- \gamma_1 \biggl(1-\frac \biggr) =2\pi\sum_^ \sin\frac \cdot\ln\Gamma \biggl(\frac \biggr) -\pi(\gamma+\ln2\pi n)\cot\frac$
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.V. Adamchik. ''A class of logarithmic integrals.'' Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997. However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846. And vol. 151, pp. 276-277, 2015.

* Rational arguments theorem: the first generalized Stieltjes constant at rational argument may be evaluated in a quasi-closed form via the following formula
:
\begin
\displaystyle
\gamma_1 \biggl(\frac \biggr)
=& \displaystyle
\gamma_1 +\gamma^2 + \gamma\ln2\pi m + \ln2\pi\cdot\ln+\frac(\ln m)^2
+ (\gamma+\ln2\pi m)\cdot\Psi\left(\frac\right) \\ mm\displaystyle & \displaystyle\qquad
+\pi\sum_^ \sin\frac \cdot\ln\Gamma \biggl(\frac \biggr)
+ \sum_^ \cos\frac\cdot\zeta''\left(0,\frac\right)
\end\,,\qquad\quad r=1, 2, 3,\ldots, m-1\,.

see Blagouchine. An alternative proof was later proposed by CoffeyMark W. Coffey ''Functional equations for the Stieltjes constants'', and several other authors.

* Finite summations: there are numerous summation formulae for the first generalized Stieltjes constants. For example,
:
\begin
\displaystyle
\sum_^ \gamma_1\left( a+\frac \right) =
m\ln\cdot\Psi(am) - \frac(\ln m)^2 + m\gamma_1(am)\,,\qquad a\in\mathbb\\ mm\displaystyle
\sum_^ \gamma_1\left(\frac \right) =
(m-1)\gamma_1 - m\gamma\ln - \frac(\ln m)^2 \\ mm\displaystyle
\sum_^ (-1)^r \gamma_1 \biggl(\frac \biggr)
= -\gamma_1+m(2\gamma+\ln2+2\ln m)\ln2\\ mm\displaystyle
\sum_^ (-1)^r \gamma_1\biggl(\frac \biggr)
= m\left\\\ mm\displaystyle
\sum_^ \gamma_1 \biggl(\frac\biggr)
\cdot\cos\dfrac = -\gamma_1 + m(\gamma+\ln2\pi m)
\ln\left(2\sin\frac\right)
+\frac
\left\\,, \qquad k=1,2,\ldots,m-1 \\ mm\displaystyle
\sum_^ \gamma_1\biggl(\frac \biggr)
\cdot\sin\dfrac =\frac (\gamma+\ln2\pi m)(2k-m)
- \frac \left\
+ m\pi\ln\Gamma \biggl(\frac \biggr) \,, \qquad k=1,2,\ldots,m-1 \\ mm\displaystyle
\sum_^ \gamma_1 \biggl(\frac \biggr)\cdot\cot\frac = \displaystyle
\frac \Big\
-2\pi\sum_^ l\cdot\ln\Gamma\left( \frac\right) \\ mm\displaystyle
\sum_^ \frac \cdot\gamma_1 \biggl(\frac \biggr) =
\frac\left\
-\frac(\gamma+\ln2\pi m) \sum_^ l\cdot \cot\frac
-\frac \sum_^ \cot\frac \cdot\ln\Gamma\biggl(\frac \biggr)
\end

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,
:$\gamma_1\left(\frac\right) = - 2\gamma\ln 2 - (\ln 2)^2 + \gamma_1 = -1.353459680\ldots$
At points 1/4, 3/4 and 1/3, values of first generalized Stieltjes constants were independently obtained by ConnonDonal F. Connon ''The difference between two Stieltjes constants''
arXiv:0906.0277
/ref> and BlagouchineIaroslav 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>
:
\begin
\displaystyle
\gamma_1\left(\frac\right) = 2\pi\ln\Gamma\left(\frac \right)
- \frac\ln\pi - \frac(\ln 2)^2 - (3\gamma+2\pi)\ln2 - \frac+\gamma_1 = -5.518076350\ldots \\ mm\displaystyle
\gamma_1\left(\frac \right) = -2\pi\ln\Gamma\left(\frac \right)
+ \frac\ln\pi - \frac(\ln 2)^2 - (3\gamma-2\pi)\ln2 + \frac+\gamma_1 = -0.3912989024\ldots \\ mm\displaystyle
\gamma_1\left(\frac \right) = -\frac\ln3 - \frac(\ln 3)^2
+ \frac\left\
+ \gamma_1 = -3.259557515\ldots
\end

At points 2/3, 1/6 and 5/6
:
\begin
\displaystyle
\gamma_1\left(\frac \right) = -\frac\ln3 - \frac(\ln 3)^2
- \frac\left\
+ \gamma_1 = -0.5989062842\ldots \\ mm\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) \\ mm\displaystyle\qquad\qquad\quad
- \frac\left\ + \gamma_1 = -10.74258252\ldots\\ mm\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) \\ mm\displaystyle\qquad\qquad\quad
+ \frac\left\+ \gamma_1 = -0.2461690038\ldots
\end

These values were calculated by Blagouchine. To the same author are also due
:
\begin
\displaystyle
\gamma_1\biggl(\frac \biggr)=& \displaystyle
\gamma_1 + \frac\left\
+ \frac \ln\Gamma \biggl(\frac \biggr)
\\ mm& \displaystyle
+ \frac \ln\Gamma \biggl(\frac \biggr)
+\left\\cdot\gamma \\ mm& \displaystyle
- \frac\left\\cdot\ln\big(1+\sqrt)
+\frac(\ln 2)^2 + \frac(\ln 5)^2 \\ mm& \displaystyle
+\frac\ln2\cdot\ln5 + \frac\ln2\cdot\ln\pi+\frac\ln5\cdot\ln\pi
- \frac\ln2\\ mm& \displaystyle
- \frac\ln5
- \frac\ln\pi\\ mm& \displaystyle
= -8.030205511\ldots \\ mm\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)
\\ mm& \displaystyle
-\left\\cdot\gamma
- \frac\big(\pi+8\ln2+2\ln\pi\big)\cdot\ln\big(1+\sqrt)
\\ mm& \displaystyle
- \frac(\ln 2)^2 + \frac\ln2\cdot\ln\pi
-\frac\ln2
-\frac\ln\pi\\ mm& \displaystyle
= -16.64171976\ldots \\ mm\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)
\\ mm& \displaystyle
-\left\\cdot\gamma
\\ mm& \displaystyle
- 2\sqrt3\big(3\ln2+\ln3 +\ln\pi\big)\cdot\ln\big(1+\sqrt)
- \frac(\ln 2)^2 - \frac(\ln 3)^2 \\ mm& \displaystyle
+ \frac\ln3\cdot\ln2
+ \sqrt3\ln2\cdot\ln\pi
-\frac\ln2 \\ mm& \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
\displaystyle
\gamma_2 \biggl(\frac \biggr) =
\gamma_2 + \frac\sum_^
\cos\frac \cdot\zeta\left(0,\frac\right) -
2 (\gamma+\ln2\pi m) \sum_^
\cos\frac \cdot\zeta''\left(0,\frac\right) \\ mm\displaystyle \quad
+ \pi\sum_^
\sin\frac \cdot\zeta''\left(0,\frac\right)
-2\pi(\gamma+\ln2\pi m)
\sum_^
\sin\frac \cdot\ln\Gamma \biggl(\frac \biggr)
- 2\gamma_1 \ln \\ mm\displaystyle\quad
- \gamma^3
-\left \gamma+\ln2\pi m)^2-\frac\rightcdot
\Psi\biggl(\frac \biggr) +
\frac\cot\frac
-\gamma^2\ln\big(4\pi^2 m^3\big) +\frac(\gamma+\ln) \\ mm\displaystyle\quad
- \gamma\big((\ln 2\pi)^2 + 4\ln m \cdot\ln 2\pi + 2(\ln m)^2\big)
- \left\\ln m
\end\,,\qquad\quad r=1, 2, 3,\ldots, m-1.

see Blagouchine.
An equivalent result was later obtained by Coffey by another method.

References

{{Reflist, 30em

Zeta and L-functions
Mathematical constants