StieltjesGamma
gives the Stieltjes constant 
.
StieltjesGamma[n,a]
gives the generalized Stieltjes constant 
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
is the coefficient of 
in the Laurent expansion of 
about the point 
. - The
are generalizations of Euler's constant; 
. 
is the coefficient of 
in the Laurent expansion of 
about the point 
. - For certain special arguments, StieltjesGamma automatically evaluates to exact values.
- StieltjesGamma can be evaluated to arbitrary numerical precision.
- StieltjesGamma automatically threads over lists.
- StieltjesGamma can be used with Interval and CenteredInterval objects. »
Examples
open all close allBasic Examples (3)
N[StieltjesGamma[1], 50]Plot values of StieltjesGamma:
ListPlot[Table[StieltjesGamma[n], {n, 0, 25}]]Evaluate a generalized Stieltjes constant to high precision:
N[StieltjesGamma[2, 1 / 2], 50]Scope (5)
TraditionalForm formatting:
StieltjesGamma[n]//TraditionalFormStieltjesGamma[n, a]//TraditionalFormEvaluate for complex second argument:
StieltjesGamma[2, 1.0 + I]The precision of the output tracks the precision of the input:
StieltjesGamma[2, 1.500000000000000000000000]StieltjesGamma[2, 1.50000000000000000000000000000000000000]StieltjesGamma threads element-wise over lists:
StieltjesGamma[{1, 2, 3}, 0.5]StieltjesGamma can be used with Interval and CenteredInterval objects:
StieltjesGamma[2, Interval[{2.34, 2.35}]]StieltjesGamma[2, CenteredInterval[3, 1 / 100]]Applications (3)
Expansion of the Riemann zeta function:
Series[Zeta[s], {s, 1, 4}]Expansion of the Hurwitz zeta function:
Series[HurwitzZeta[s, a], {s, 1, 3}]Test Li’s criterion for the Riemann hypothesis:
Limit[Table[1 / (n - 1)!D[ s^n - 1 Log[(s (s - 1) Gamma[(s/2)] Zeta[s]/2 π^s / 2)], {s, n}], {n, 3}], s -> 1] //FullSimplify//TraditionalFormAll values should be positive:
N[%]Express integrals in terms of StieltjesGamma:
Integrate[(x^1 / 4Log[Log[1 / x]]/(1 + x)), {x, 0, 1}]Properties & Relations (2)
The EulerGamma case evaluates automatically:
StieltjesGamma[0]StieltjesGamma[0, a]Various symbolic relations are automatically used:
0 < StieltjesGamma[10] < 10 ^ -3Table[Abs[StieltjesGamma[n]] < 2(n - 1)! / Pi ^ n, {n, 5}]Possible Issues (4)
Substitution of derivatives of Zeta at 
yields indeterminate values:
Derivative[3][(Zeta[#] + 1 / (1 - #))&][1]
Use Limit to obtain the expansion coefficient:
Limit[Derivative[3][(Zeta[#] + 1 / (1 - #))&][s], s -> 1]The argument of StieltjesGamma must be an exact non-negative integer:
StieltjesGamma[2.]
Use N to obtain a numerical approximation:
N[StieltjesGamma[2]]Alternatively, use two-argument form:
StieltjesGamma[2, 1.0]StieltjesGamma does not allow numericalization of its index:
N[StieltjesGamma[2n]]N[StieltjesGamma[2n, 3 / 2]]It is currently not known if Stieltjes constants are algebraic numbers:
Element[StieltjesGamma[2], Algebraics]Related Links
MathWorld
The Wolfram Functions Site
NKS|OnlineText
Wolfram Research (1996), StieltjesGamma, Wolfram Language function, https://reference.wolfram.com/language/ref/StieltjesGamma.html (updated 2022).
CMS
Wolfram Language. 1996. "StieltjesGamma." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/StieltjesGamma.html.
APA
Wolfram Language. (1996). StieltjesGamma. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StieltjesGamma.html