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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.

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  • $\begingroup$ "holomorphic for Re$(s) > 1/k$ with possible poles at $1,0,-1,\dots$" doesn't make sense: if a pole at $s=1$ is possible then it could be meromorphic but not holomorphic; and talking about poles at $0,-1,\dots$ isn't reasonable if the function isn't even defined there. Please clarify. $\endgroup$ – Greg Martin Mar 16 '18 at 18:14
  • $\begingroup$ What is $\|\delta_j^i\|$, i.e. what is the symbol $\|\cdot\|$? $\endgroup$ – GH from MO Mar 16 '18 at 19:34
  • $\begingroup$ I intended the absolute value. If they are not uniformly bounded the result should fail. $\endgroup$ – user39115 Mar 16 '18 at 19:48
  • $\begingroup$ For the first question to make sense, you need to know that $Z(s)$ extends meromorphically to the left of $1/k$. How do you know this? When all $P_i$s are equal, it is known, but in general? $\endgroup$ – A Stasinski Mar 17 '18 at 18:50
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    $\begingroup$ If I remember correctly, Pierrette Cassou-Nogues has several papers on the subject. $\endgroup$ – Henri Cohen Mar 20 '18 at 9:44

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