5 problems

Hardy–Littlewood conjecture. If $\alpha$ is 1-Diophantine but not quadratic irrational, then the line

Let $W$ be a type III polynomial: it is as in type II, and there are infinitely many pairs $n,m$ with $c_{n,m}>0$ and $\frac{n+1}{m}>\beta$. These polynomials fall under case (3).…

Let $W(x,y)=\sum_{n,m}a_{n,m}x^ny^m$ be an integral polynomial with $W(x,0)=1$, and define … If $\beta=\max\{\frac{n}{m}:m\geq 1,\ a_{n,m}\neq 0\}$, then the du Sautoy–Woodward con…

Let $W(x,y)=\sum_{n,m}a_{n,m}x^ny^m$ be an integral polynomial with $W(x,0)=1$, and define … If $\beta=\max\{n/m:a_{n,m}\neq 0\}$, then the Euler-product continuation conjecture.…

Natural-boundary conjecture. The analytic continuation of $\mathcal{Z}_3(s)\equiv\mathcal{M}_6(s)$ has