I was interested in an arithmetic function satisfying a certain property, I am not sure at the moment if such thing even exists or not. But I was wondering maybe I could get some hint or idea or input from someone.

Fix $k \in \mathbb{N}$ and some $\epsilon > 0$. I would like a function $f$ defined for all $\mathbb{Z}$ such that
$$| \ f(n_1) ... f(n_k) \ | \ll | \ n_1+ n_2 + .. + n_k \ |^{-\epsilon} $$

and
$$| \ f(n_1) ... f(n_k) \ | \gg 1 ~~~\text{if}~~\ n_1+ n_2 + .. + n_k = 0$$

I would greatly appreciate any assistance on this matter! Thank you very much!