As my post history will tell you, I am not particularly good at doing things practically miles above my level. Having said that, I'm using Desmos to look at values of $\frac{\pi^{2n-1}}{\zeta(2n-1)}$ (using the closest thing to the Reimann-Zeta function I can get in Desmos) to see if it produced integral results similarly to $\zeta(2n)$. Then this happened:
Why is this happening?
On
Your code looks at the fractional part of $\frac{\pi^{2n-1}}{\zeta(2n-1)}$.
My bet is that at some point when $n$ becomes large, the number is so large that the computer simply neglects the fractional part.
Edit: when $n$ is large, the denominator is very close to $1$, and the numerator increases exponentially. If you ask Desmos the fractional part of $\pi^N$, you get non-zero answers until $N=2\times 16-1$ where the number is so large that Desmos apparently forgets about the fractional part. There may be a trick to force him remember it.
I guess the calculator uses IEEE-double precision. As noted by @arnaud-mortier you plot the fractional part of $\frac{\pi^k}{\zeta(k)}.$
If you compute powers of $\pi$ the exponent of the double precision number $\pi^k$ increases and the available bits for the fractional part is decreasing. Here is the table for $\pi^k$
$$\frac{\pi^{31}}{2^{52}} \approx 0.5729 \quad \frac{\pi^{32}}{2^{52}} \approx 1.7998$$ and a table for $\frac{\pi^k}{\zeta(k)}$
The table are nearly identical for larger $k$ because $\zeta(k)$ rapidly converges to $1$.
To summarize: all IEEE double precision numbers $> 2^{52}$ are rounded to integers and have no fractional part.