HP Prime (emulator): Zeta function query

[Paul Townsend (UK)]

The HP Prime emulator seems to have difficulty with Zeta() of an odd positive integer, retaining the function call unevaluated even after Eval(). In a plot view such values show as "Undefined". What's going on? Zeta() is well-defined at these values and my WP34s (the real thing) has no difficulty evaluating Zeta(3) for example to be 1.20205...

Zeta() of an even integer is correctly handled returning the appropriate expression involving pi, and Zeta() of non-integral values is also correctly handled.

[Helge Gabert]

In CAS mode, hit approx key, after the Prime returns Zeta(3), then you also get the desired numerical result (1.202...).

I agree, in plot view you get "undefined".

[Paul Townsend (UK)]

I wonder if it is evaluating Zeta(n) recursively using Zeta(n-2)? Starting with Zetas(3) would result in the algorithm boggling on Zeta(1) and bombing out with an error.

Asking for Zeta(1) correctly returns infinity.

[Helge Gabert]

Yes, I bet that's it! The algorithm should change depending on the user input - - but since this "only" affects odd integers > 1 it was probably either overlooked or deemed to be too much trouble to implement.

[Paul Dale]

Sounds like this is related to zeta(2n) being expressible relatively simply in terms of PI and the Bernoulli numbers whereas zeta(2n+1) are not.

Zeta(3) is Apéry's constant which has been proven to be transcendental, there are infinite series expressions for it but nothing really simple and certainly nothing simpler than just zeta(3).

- Pauli

[Gerson W. Barbosa]

That's what I thought, but that works for non-integer values and this is defined for integers only.

http://people.math.sfu.ca/~cbm/aands/page_807.htm

(Abramowitz and Stegun: Handbook of Mathematical Functions, page 807, formula 23.2.16)

Gerson.

P.S.: This HP 50g program has the same behaviour regarding integer arguments.

%%HP: T(3)A(D)F(.);

\<< -22 CF \-> n

  \<< '(2.*\pi)^n/(2.*n!)' \->NUM n IBERNOULLI ABS INV /

  \>>

\>>

Edited: 29 Sept 2013, 6:54 p.m.

[mjcohen]

I don't think it has been proven transcendental - only irrational.

[Paul Dale]

You are correct, the proof was for irrationality only.

- Pauli