So we're all sprickety the same lingity...

You're asking for a proof or counterexample for the following:

Define $\displaystyle f_p:\mathbb{N} \rightarrow\mathbb{N} $ by $\displaystyle f_p(s)=\sum_{n=1}^{p-2} n^s$ for some prime $\displaystyle p$

Theorem:

(i) $\displaystyle f_p(s) \equiv 0 (\bmod p)$ when $\displaystyle s$ is even

(ii) $\displaystyle f_p(s) \equiv -1 (\bmod p)$ when $\displaystyle s$ is odd

(iii) $\displaystyle f_p(s) \equiv -2 (\bmod p)$ when $\displaystyle s$ is of the form $\displaystyle k(p-1)$ for some integer $\displaystyle k > 1$

Is this correct? ***I think I may be misinterpreting you, as this theorem as I have written it is very wrong.