Thanks Brad.

Presumably, your ratio of zeta functions provide an upper bound as $n \rightarrow \infty$. I had a quick look at Polya and Szego but did not manage to find it.

My actual sum is $n^k S(n,k)$ and the bound above would be $O(n^k)$ but I'd like a tighter upper bound, say $O(n^{k-1} \log(n))$ or something like this, if at all possible.

I am considering signal expansions along these kinds of regularly sampled support sets, and correlation moments between non-orthogonal expansion coefficients.

If we let $n=5,$ and decompose $(f(1),\ldots,f(5))$ by taking inner products with the vectors $\{(1,1,1,1,1),(0,1,0,1,0),(0,0,1,0,0),\ldots,(0,\ldots,0,1)\}$ the correlations between these expansion coefficients lead directly to $n^k S(n,k).$