Contradicting statements about the Riemann zeta function at positive odd integers

Question

I have found two contradicting statements about the value of $\zeta(k)$ when $k=2n+1$ and $n\in\mathbb{Z_0^+}$. Which one is correct?

1. "The Riemann zeta function for odd integers has no known closed-form formula", that is, you can't find $\zeta(k)=f(k)$ such that $f$ holds for all positive odd integers. This is apparently a "well-known" fact.

2. Wikipedia states here that $\zeta(k)$ can be expressed for all positive integers, even or odd, using the formula:

   $\zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)}\quad(k=2,3,\dots)$

which involves primorials and the Jordan totient function.

EDIT: I understand the above is not a closed formula. But then what about this paper, which proposes an explicit closed-form formula?

Answer 1

The paper you link to proposes "closed forms" of the type

$$\zeta(3)=\frac{2^3}{2^3-1}\beta(3)-\dots$$

with

$$\beta(s)=\sum_{k=1}^\infty\frac{(-1)^{k-1}}{(2k-1)^s}$$

Thus the infinite term is just hidden away in the function $\beta$ (which occurs in every expression they give) and the claim of having obtained closed forms is a tall one indeed.