Built-in Mathematica Symbol

StieltjesGamma

StieltjesGamma[n]
gives the Stieltjes constant

.

StieltjesGamma[n, a]
gives the generalized Stieltjes constant

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (3)

Evaluate to high precision:

Plot values of StieltjesGamma:

Evaluate to high precision:

In[1]:=

Out[1]=

Plot values of StieltjesGamma:

Scope (4)

TraditionalForm formatting:

Evaluate for complex second argument:

The precision of the output tracks the precision of the input:

StieltjesGamma threads element-wise over lists:

Applications (3)

Expansion of the Riemann zeta function:

Expansion of the Hurwitz zeta function:

Test Li's criterion for the Riemann hypothesis:

All values should be positive:

Express integrals in terms of StieltjesGamma:

Properties & Relations (2)

The EulerGamma case evaluates automatically:

Various symbolic relations are automatically used:

Possible Issues (4)

Substitution of derivatives of Zeta at

yields indeterminate values:

Use Limit to obtain the expansion coefficient:

The argument of StieltjesGamma must be an exact non-negative integer:

Use N to obtain a numerical approximation:

Alternatively, use two argument form:

StieltjesGamma does not allow numericalization of its index:

It is currently not known if Stieltjes constants are algebraic numbers: