Weakly Divisible Rings

We define a new class of rings parameterized by binary forms of a certain type, and give an effective lower bound for the number of such rings whose discriminant is less than a bound $X$. We also obtain a lower bound for the number of number fields whose ring of integers lies in the above class and whose discriminant is less than a bound $X$. Our results improve an estimate of Bhargava-Shankar-Wang in \cite{bhargava2022squarefree}. In particular we show the following: $\bullet$ When $n\ge 4,$ the number of rings of rank $n$ over $\mathbb{Z}$ with discriminant less than or equal to $X$ is $$\gg_n X^{\frac{1}{2}+\frac{1}{n-\frac{4}{3}}}.$$ $\bullet$ When $n\ge 6,$ the number of number fields of degree $n$ with discriminant less than $X$ is $$\gg_{n,\epsilon} X^{\frac{1}{2} +\frac{1}{n-1} + \frac{(n-3)r_n}{(n-2)(n-1)}-\epsilon}$$ where $r_n=\frac{\eta_n}{n^2-4n+3-2\eta_n (n+\frac{2}{n-2})}$ and where $\eta_n$ is $\frac{1}{5n}$ if $n$ is odd and is $\frac{1}{88n^6}$ when $n$ is even.