Weakly Divisible Rings

Abstract 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 is in the above class and whose discriminant is less than a bound $X$. Our results improve the estimate of Bhargava–Shankar–Wang in [7]. In particular, we show the following: •When $n\ge 4,$ the number of rings of rank $n$ over $\mathbb{Z}$ with discriminant less than or equal to $X$ is $$ \begin{align*} & \gg_n X^{\frac{1}{2}+\frac{1}{n-\frac{4}{3}}}. \end{align*} $$•When $n\ge 6,$ the number of number fields of degree $n$ with discriminant less than $X$ is $$ \begin{align*} & \gg_{n,\epsilon} X^{\frac{1}{2} +\frac{1}{n-1} + \frac{(n-3)r_n}{(n-2)(n-1)}-\epsilon}, \end{align*} $$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.

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