Exact bounds for even vanishing of $K_* (\mathbb{Z}/p^n)$

In this note, we prove that $K_{2i} (\mathbb{Z}/p^n) \neq 0$ if and only if $p-1$ divides $i$ and $0 \leq i \leq (p-1) p^{n-2}$, refining the even vanishing theorem of Antieau, Nikolaus and the first author in this case. As a corollary of our proof, we determine that the nilpotence order of $v_1$ in $\pi_* K(\mathbb{Z}/p^n)/p$ is equal to $\frac{p^n-1}{p-1}$. Our proof combines the recent crystallinity result for reduced syntomic cohomology of Hahn, Levy and the second author with the explicit complex computing the syntomic cohomology of $\mathcal{O}_K /\varpi^n$ constructed by Antieau, Nikolaus and the first author.