On a conjecture of Ramírez Alfonsín and Skałba III

Let $1<c<d$ be two relatively prime integers. For a non-negative integer $\mathrm{\ell }$, let $g_{\mathrm{\ell }}(c,d)$ be the largest integer $n$ such that $n=cx+dy$ has at most $\mathrm{\ell }$ non-negative solutions $(x,y)$. In this paper we prove that $${\pi }_{\mathrm{\ell },c,d}\sim \frac{\pi (g_{\mathrm{\ell }}(c,d))}{2\mathrm{\ell }+2}(\text{as }c\to \mathrm{\infty }),$$ where ${\pi }_{\mathrm{\ell },c,d}$ is the number of primes $n$ having more than $\mathrm{\ell }$ distinct non-negative solutions to $n=cx+dy$ with $n\le g_{\mathrm{\ell }}(c,d)$, and $\pi (x)$ denotes the number of all primes up to $x$. The case where $\mathrm{\ell }=0$ has been proved by Ding, Zhai and Zhao recently, which was conjectured formerly by Ramírez Alfonsín and Skałba.