Natural Boundaries of Polynomial Euler Products

Definitions and mathematical statement

Let $W(x,y)=\sum a_{n,m}x^ny^m\in{\mathbb{Z}}[x,y]$ satisfy $W(x,0)=1$, and define, initially in a right half-plane, \[ D(s)=\prod_{p\text{ prime}}W(p,p^{-s}). \] For nonconstant $W$, retain the source's exact number \[ \beta=\max\left\{\frac{n+1}{m}:a_{n,m}\ne0,\ m>0\right\}. \] The product converges absolutely for $\operatorname{Re}s>\beta$. The case $W=1$ gives $D=1$ and an empty maximum; it belongs to the trivial product case rather than a proposed boundary case.

The first question is whether $D$ continues meromorphically to all of ${\mathbb{C}}$ exactly when it is a finite product of Riemann zeta-functions. The second proposed assertion says that otherwise $\operatorname{Re}s=\beta$ is a natural boundary: no point of that line permits local meromorphic continuation of the function from its right-hand domain.

"Finite product" needs clarification. Polynomial Euler factors naturally produce integer powers and shifted/scaled arguments such as $\prod_{j=1}^{r}\zeta(a_js+b_j)^{e_j}$, including negative exponents. Already $W=1-y$ gives $D(s)=1/\zeta(s)$, so products of unshifted zeta-functions with only positive exponents are not an adequate default interpretation. The allowed arguments, exponents and finite local factors are not specified.

The displayed $\beta$ is an absolute-convergence bound and cannot simply be identified with a continuation boundary. For example, for $W=1-2y$ it equals $1$, while \[ D(s)=\zeta(s)^{-2}\prod_p \frac{1-2p^{-s}}{(1-p^{-s})^2} \] gives a meromorphic continuation to $\operatorname{Re}s>1/2$: the remaining local factors are $1+O(p^{-2\operatorname{Re}s})$ away from finitely many small primes. Thus this example crosses the proposed line. The boundary formula and the precise zeta-product exception must be checked against the intended source statement; the printed $(n+1)/m$ is not silently replaced by a different slope.

Short English statement

Classify which polynomial Euler products extend meromorphically to the whole plane, and locate a natural boundary otherwise. The finite-zeta-product convention is undefined, and the printed slope is an absolute-convergence threshold that need not be a natural boundary.

Sources

• Supplied report: Profinite and Asymptotic Group Theory (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/28, pp. 1537--1588; proposed year 2008. https://doi.org/10.4171/owr/2008/28.
• Supplied du Sautoy--Woodward monograph: https://doi.org/10.1007/978-3-540-74701-7.
• Local UnsolvedMath registry, permanent ID 30000988, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note reports comparison with the report. The supplied 21 August 2026 literature triage labels the record partially solved, reporting substantial natural-boundary criteria while treating the full iff classification as conjectural. These dated claims are retained alongside the printed-slope issue, without an independent literature assessment.