Quadratic Class Numbers Divisible by 3

Abstract

Let $N_+(X)$ denote the number of distinct real quadratic fields $\mathbb{Q}(\sqrt{d})$ with $d\leq X$ for which $3|h(\mathbb{Q}(\sqrt{d}))$. Define $N_-(X)$ similarly for $\mathbb{Q}(\sqrt{-d})$. It is shown that $N_+(X), N_-(X)\gg X^{9/10-\varepsilon}$ for any $\varepsilon>0$. This improves results of Byeon and Koh [2] and of Soundararajan [7], which had exponent $7/8-\varepsilon$.