Title: Primes of the form $ax+by$ in certain intervals with small solutions Show Chinese title

Title: 形如$ax+by$的素数在某些区间内具有小解

Abstract: Let $1<a<b$ be two relatively prime integers and $\mathbb{Z}_{\ge 0}$ the set of non-negative integers. For any non-negative integer $\ell$, denote by $g_{\ell,a,b}$ the largest integer $n$ such that the equation $$n=ax+by,\quad (x,y)\in\mathbb{Z}_{\ge 0}^{2} \quad (1)$$ has at most $\ell$ solutions. Let $\pi_{\ell,a,b}$ be the number of primes $p\leq g_{\ell,a,b}$ having at least $\ell+1$ solutions for (1) and $\pi(x)$ the number of primes not exceeding $x$. In this article, we prove that for a fixed integer $a\ge 3$ with $\gcd(a,b)=1$, $$ \pi_{\ell,a,b}=\left(\frac{a-2}{2(\ell a+a-1)}+o(1)\right)\pi\bigl(g_{\ell,a,b}\bigr)\quad(\text{as}~ b\to\infty). $$ For any non-negative $\ell$ and relatively prime integers $a,b$, satisfying $e^{\ell+1}\leq a<b$, we show that \begin{equation*} \pi_{\ell,a,b}>0.005\cdot \frac{1}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} Let $\pi_{\ell,a,b}^{*}$ be the number of primes $p\leq g_{\ell,a,b}$ having at most $\ell$ solutions for (1). For an integer $a\ge 3$ and a large sufficiently integer $b$ with $\gcd(a,b)=1$, we also prove that $$ \pi^{*}_{\ell,a,b}>\frac{(2\ell+1)a}{2(\ell a+a-1)}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. $$ Moreover, if $\ell<a<b$ with $\gcd(a,b)=1$, then we have \begin{equation*} \pi^{*}_{\ell,a,b}>\frac{\ell+0.02}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} These results generalize the previous ones of Chen and Zhu (2025), who established the results for the case $\ell=0$. Show Chinese abstract

Abstract: 设$1<a<b$为两个互质的整数，$\mathbb{Z}_{\ge 0}$为非负整数的集合。 对于任何非负整数$\ell$，记$g_{\ell,a,b}$为最大的整数$n$，使得方程$$n=ax+by,\quad (x,y)\in\mathbb{Z}_{\ge 0}^{2} \quad (1)$$最多有$\ell$个解。 设$\pi_{\ell,a,b}$为满足方程(1)至少有$\ell+1$个解的素数$p\leq g_{\ell,a,b}$的个数，$\pi(x)$为不超过$x$的素数的个数。 在本文中，我们证明对于一个固定的整数$a\ge 3$且$\gcd(a,b)=1$，$$ \pi_{\ell,a,b}=\left(\frac{a-2}{2(\ell a+a-1)}+o(1)\right)\pi\bigl(g_{\ell,a,b}\bigr)\quad(\text{as}~ b\to\infty). $$。对于任何非负的$\ell$和互质的整数$a,b$，满足$e^{\ell+1}\leq a<b$，我们证明\begin{equation*} \pi_{\ell,a,b}>0.005\cdot \frac{1}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*}。设$\pi_{\ell,a,b}^{*}$是满足 (1) 的解最多为$\ell$的素数$p\leq g_{\ell,a,b}$的个数。 对于整数$a\ge 3$和足够大的整数$b$满足$\gcd(a,b)=1$，我们还证明了$$ \pi^{*}_{\ell,a,b}>\frac{(2\ell+1)a}{2(\ell a+a-1)}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. $$。此外，如果$\ell<a<b$满足$\gcd(a,b)=1$，那么我们有\begin{equation*} \pi^{*}_{\ell,a,b}>\frac{\ell+0.02}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*}这些结果推广了 Chen 和 Zhu（2025）之前的成果，他们建立了当$\ell=0$时的结果。