Yesterday, I was looking for references about meromorphic extensions of certain Dirichlet associated with polynomials. I was surprised to discover that much less than I previously thought was known.
I can now ask two very specific questions regarding the meromorphic extensions of certain Dirichlet series:
Let $P_i(x)=\prod_{j=1}^k(x+\delta_j^i)$ be a family of polynomial indexed by $i=1,2\ldots,$ with real coefficients and $\mathcal{Re}(\delta_j^i)>-1$ for $j=1, \ldots k.$ Assume also that there exists $0<M<\infty$ such that the absolute values $|\delta_j^i|<M$ for every $i,j.$
Is the associated Zeta $$ Z(s)=\sum_{n=1}^{\infty}\frac{1}{P_n(n)^s}$$ holomorphic for $\mathcal{Re}(s)>\frac{1}{k}$ and it has analytic continuation in the whole complex plane with only possible poles at $\frac{j}{k}$ for $j=1,0,-1,-2,\ldots$ other than non-positive integers?
I thought this was obviously true, but I was unable to prove it, to disprove it, and not even able to find any proof.
The cases where $P_i=P_j$ for every $i,j$ is in M. Eie, On a Dirichlet series associated with a polynomial, Proc. Amer. Math. Soc. 110 (1990), no. 3, 583–590.
My second question is the following.
Assuming that $P=P_i=P_j$ for every $i,j.$ Under which ''general'' hypothesis does $Z$ have a pole at $1/k$ and not have a pole at $0?$
It is well known that for $P(n)=n+q,$ we recover the Hurwitz Zeta function, that can be meromorphically extended in the whole complex plane with a simple pole at $1.$ I would be quite surprised if no other more general examples are known to have a simple pole at $1/deg(P)$ and not at $0,$ by looking at the Dirichlet series associated with polynomials.
