I'm trying to gain a better understanding of the difficulties establishing an exact form for $\zeta(2n+1)$
Taking $\zeta(3)$ as a case in point, I can construct the Euler product over primes:
I can see that when the powers in such products are all even, they behave nicely but in this case (and other odd $s$ the product contains odd powers, making the objective difficult for us. Intuitively for me, this is because the known values for $\zeta(n)$ are closed to even numbers and the values at odd $n$ are in a sense transcendental. Notionally for me, this "closedness" is related to the fact that $\zeta(1)$ diverges so there's perhaps a morphism to some algebra in which $1$ is transcendental.
I have the idea that since $\zeta(2n)$ converges as $n\to\infty$ we need to metrize the distance $\lvert\zeta(2n)-\zeta(n)\rvert$ which goes to zero as $n\to\infty$ and create space not dissimilar to $2-$adics in which we then have something of utility in measuring the distance $\lvert\zeta(n+1)-\zeta(n)\rvert$.
Is this a pre-existing approach and are there any pointers where I might see how this is done?