Ramírez Alfonsín–Skałba conjecture on primes represented by two generators

Let $1<c<d$ be relatively prime integers and set

$$g_{c,d}=cd-c-d.$$

Let $\pi_{c,d}$ denote the number of primes not exceeding $g_{c,d}$ that can be written as $cx+dy$ with $x,y\in\mathbb{Z}_{\geqslant0}$, and let $\pi(t)$ denote the number of primes up to $t$. Ramírez Alfonsín–Skałba conjecture. One has

$$\pi_{c,d}\sim\frac{\pi(g_{c,d})}{2}\quad (\text{as }c\rightarrow\infty).$$

The conjecture predicts that asymptotically half of the primes up to the two-generator Frobenius number are representable in this way. The paper proves the conjecture using the Hardy–Littlewood method; it had previously been established for almost all admissible pairs $c,d$.