12 problems

Let $P\in \mathbb{Z}[t]$ be irreducible, and let $\mathfrak S_P$ denote the singular series … An integer with exactly two distinct prime divisors, called an $E_2$ integer, has the…

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

Bateman–Horn conjecture. With

Uniform Bateman–Horn conjecture. One has

Let $k$ be even, let $\pi_k(x)$ count primes $p\leq x$ for which $p+k$ is prime, and define … Bateman--Horn prediction for fixed prime gaps. … This refines Polignac's infinitude as…

Let $f_1(t)=t$ and $f_2(t)=t+2$, and let $Q(f_1,f_2;x)$ count $n\leq x$ for which both values are prime. Define the twin-prime constant … Bateman--Horn prediction for twin primes.…

Let $f_1,\ldots,f_k\in\mathbb Z[x]$ be irreducible polynomials with positive leading coefficients, no two multiples of one another, and let $f=f_1\cdots f_k$ not vanish identically…

Let $f_1,f_2,\ldots,f_k\in\mathbb Z[x]$ be distinct irreducible polynomials with positive leading coefficients. Let $f=f_1f_2\cdots f_k$, assume that $f$ does not vanish identicall…

Let $F_1,\ldots,F_m\in\mathbb{F}_q[t][x]$ be non-associate irreducible polynomials that are separable over $\mathbb{F}_q(t)$, and let $n\geq 3$. The monicity condition on the $F_i$…

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