Generalized Riemann zeta function

[axkr]

Is the "generalized Riemann zeta function" zeta(s,a) also implemented?

The generalized Riemann zeta function is identical to the hurwitz zeta function for Re(a)>0

for Re(a)<0 it is defined as: $$\sum_{k=0}^{\infty} { ((k+a)^2)^{-s/2} }$$

See:

• https://reference.wolfram.com/language/ref/Zeta.html

[mtommila]

Not sure if that definition of "generalized Riemann zeta function" is used anywhere else than in Mathematica. Do you have any other references?

[axkr]

Don't know if this counts as another reference:

• https://mathworld.wolfram.com/HurwitzZetaFunction.html

[mtommila]

Well, it's at least mentioned in some of the references there, although I'm not quite sure if the references are talking about the Hurwitz zeta function or that "generalized" zeta function

Is the "generalized zeta function" useful for something where the Hurwitz zeta function can't be used?

[mtommila]

There is apparently a formula that connects the Hurwitz zeta function and the "generalized Riemann zeta function"

https://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/02/

(Where "zeta" is the "generalized Riemann zeta" and "classical zeta" is the Hurwitz zeta)

But overall it's somewhat confusing that even on that page it claims that the "Hurwitz zeta function" has Mathematica notation Zeta[s, a] when this is not actually true.

Having gone through several of the algorithms that use ζ(s, a) on the functions.wolfram.com site, they seem to use the Hurwitz zeta function and not the "generalized Riemann zeta function" (which seems pretty bogus anyways at least due to the rather arbitrary looking rule "where any term with k + a = 0 is excluded").

[axkr]

Having gone through several of the algorithms that use ζ(s, a) on the functions.wolfram.com site, they seem to use the Hurwitz zeta function and not the "generalized Riemann zeta function" (which seems pretty bogus anyways at least due to the rather arbitrary looking rule "where any term with k + a = 0 is excluded").

I think there are historical reasons, Zeta is used in Mathematica since 1988, HurwitzZeta is introduced relatively late in 2008.