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!



Existence of arithmetic function satisfying a certain property
Suppose that $f$ satisfies your conditions: $|f(n\_1)\dotsb f(n\_k)|\le
A|n\_1+\dotsb+n\_k|^{-\epsilon}$ if $n\_1+\dotsb+n\_k\ne 0$, and
$|f(n\_1)\dotsb f(n\_k)|\ge B$ if $n\_1+\dotsb+n\_k=0$, with positive $A$ and
$B$ depending only on $k$ and $\epsilon$. The second condition implies
that $\max\{|f(n)|,|f(-n)|\}\ge \sqrt B/|f(0)|^{k/2-1}$, for every
positive integer $n$ (take $n\_1=n,n\_2=-n$, and $n\_3=\dotsb=n\_k=0$). Using
now the first condition with the $n\_i\in\{-n,n\}$ chosen to satisfy
$|f(n\_i)|\ge \sqrt B/|f(0)|^{k/2-1}$ (and all equal each other), we get
$B^{k/2}/|f(0)|^{k(k/2-1)}\le A(kn)^{-\epsilon}$. This cannot be true for
$n$ large enough. Consequently, no such function exists.


