I need at least basic information about generating functions of the following class of arithmetic functions, grouped by levels $k$.

Fix some $k$ and some family $\varepsilon_*=(\varepsilon_\sigma)_{\sigma\subseteq\{1,...,k\}}$ of integers, and let $f_{\varepsilon_*}(n)$ be the number of representations $$ n=\sum_{\sigma\subseteq\{1,...,k\}}\varepsilon_\sigma\prod_{i\in\sigma}x_i $$ where $x_i$ run through natural numbers.

Can one say anything definite about the series $\sum_{n>0}\frac{f_{\varepsilon_*}(n)}{n^s}$ or $\sum f_{\varepsilon_*}(n)q^n$ in this generality? Say, some simplifications, more explicit expressions, or expressions through series with smaller $k$, or whatever.

For example, the number $\tau_k(n)$ of representations of $n$ in the form $x_1\cdots x_k$ has Dirichlet generating function $\zeta(s)^k$, but even here I am not aware of any handy Lambert series: for $k=2$ it is $\sum_{d>0}\frac{q^d}{1-q^d}$, but already for $k=3$ I cannot do any better than $\sum_{d>0}\frac{\tau_2(d)q^d}{1-q^d}$ or $\sum_{d,e>0}\frac{q^{de}}{1-q^{de}}$ which is not so convenient for my purposes. Supposedly one has to simplify the inverse Mellin transform of $\zeta(s)^k$ but I cannot figure out how.

If this helps, I can impose quite severe restrictions on $\varepsilon_\sigma$, e. g. I can assume that $\varepsilon_\sigma\in\{-1,0,1\}$, that $\varepsilon_{\{1,...,n\}}=1$, that $\varepsilon_\sigma=0$ for $\sigma$ of cardinality $n-1$, that $\varepsilon_\varnothing=1$, or that $\varepsilon_\varnothing=0$, things like that. It is also OK to consider the case when $x_i$ are allowed to be zero.

In "typical" or "generic" cases like, say, the number of representations $$ n=xyztu-xyz+xzt+yu-y+t $$ I don't really know what to do. Ideally I would like to have some good-looking Lambert series; a Dirichlet series would be also useful, modulo my above concerns with the inverse Mellin transform.

Maybe also some entirely different approaches exist, I don't know.


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