Relation between $1-(n^{p-1}\mod p)$ and Riemann $\zeta$

Question

Taking: $$\mathcal V_p=1-(n^{p-1}\mod p)$$ with $$\lim_{p\rightarrow \infty}\mathcal V_p = \operatorname{sinc}(2\pi \,n)$$ and Riemanns' well known functional equation, I get easily to this result: $$\lim_{p\rightarrow \infty}[1-(n^{p-1}\mod p)] = \frac{\zeta(2n)}{\zeta(1-2n)}\left(\frac{1}{n\;(2\pi)^{2n}\;\Gamma(1-2n)}\right)$$

It is not spectacular, and I just beg your support, did I calculate something wrong, or is this true?

Answer 1

just to help to close the case. seems the answer to my question is clear: the calculation is correct but the result is trivial.