6 problems

Finite-field Bateman–Horn conjecture. As $X\to\infty$ through powers of $q$,

Finite-field Bateman–Horn conjecture. As $X\to\infty$ through powers of $q$,

Profinite Bateman–Horn conjecture. The following limits exist and satisfy

Let $f_1(t)=a_1t+b_1$ and $f_2(t)=a_2t+b_2$ be linear forms with integers $a_1,a_2,b_1,b_2$ and $a_1,a_2>0$. Set $f=f_1f_2$, and suppose that $f$ is admissible, meaning that it doe…

Function-field Bateman–Horn conjecture. As $X\to\infty$ through powers of $q$,

Average quadratic prime-pair Dirichlet-series conjecture. For $\sigma>1/4$ and $N\to\infty$,