Zeta
Usage Message: Zeta[s] gives the Riemann zeta function. Zeta[s, a] gives the generalized Riemann zeta function.
Attributes[Zeta] = {Listable, NumericFunction, Protected} Options: IncludeSingularTerm
-> False Related Symbols: LerchPhi
Notes: Numerical evaluation of Zeta[s, a] is based on the definition
Sum[1/((a + k)^2)^(s/2), {k, 0, Infinity}]
Most references for this function use a different definition:
Sum[1/(a + k)^s, {k, 0, Infinity}]
The definition that is used in Mathematica was chosen because it is believed to have a more convenient branch cut structure for modern uses of this function. Although this definition has been widely discussed by experts on special functions, we are not aware of a published reference for this definition. Known Bugs: The documentation for Zeta[s, a], and some functions in Mathematica, use the definition
Sum[1/(a + k)^s, {k, 0, Infinity}]
This definition is in disagreement with the definition that is used for computing numerical values of Zeta[s, a].
Additional Online Documentation:
Mathematica 3.0
http://documents.wolfram.com/v3/RefGuide/Zeta.html
Mathematica 4.0
http://documents.wolfram.com/v4/RefGuide/Zeta.html
Questions or comments? Send email to support@wolfram.com.
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