The du Sautoy–Woodward conjecture on natural boundaries of polynomial Euler products

Let $W(x,y)=\sum_{n,m}a_{n,m}x^ny^m$ be an integral polynomial with $W(x,0)=1$, and define

$$D(s)=\prod_p W(p,p^{-s}).$$

If $\beta=\max\{\frac{n}{m}:m\geq 1,\ a_{n,m}\neq 0\}$, then the du Sautoy–Woodward conjecture. $D(s)$ is meromorphically continuable to the whole complex plane if and only if it is a finite product of Riemann zeta-functions; in the latter case, the line $\Re s=\beta$ is the natural boundary of $D$.

This conjecture concerns the dichotomy between Euler products with global meromorphic continuation and those having a natural boundary. The source presents it as believed to be true; related classes of polynomials are proved to have the asserted natural-boundary behavior, in some cases conditional on the Riemann hypothesis.