Class numbers of indefinite binary quadratic forms

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 312-333

Résumé

Let $h(d)$ be the class number of properly equivalent primitive binary quadratic forms $ax^2+bxy+cy^2$ of discriminant $d=b^2-4ac$. The case of indefinite forms $(d<0)$ is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let $\alpha(x)$ be an arbitrarily slow monotonically increasing function such that $\alpha(x)\to\infty$. Then $$ \#\biggl\{p\le x\bigg\vert\biggl(\frac5p\biggr)=1,\,h(5p^2)>(\log p)^{\alpha(p)}\biggr\}=o(\pi(x)), $$ where $\pi(x)=\#\{p\le x\}$. 2. Let $F$ be an arbitrary sufficiently large positive constant. Then for $x>x_F$ , the relation $$ \#\biggl\{p\le x\bigg\vert\biggl(\frac 5p\biggr)=1,\,h(5p^2)>F\biggr\}\asymp\frac{\pi(x)}F $$ holds. 3. The relation $$ \#\biggl\{p\le x\bigg\vert\biggl(\frac5p\biggr)=1,\,h(5p^2)=2\biggr\}\sim\frac9{19}A\pi(x) $$ holds, where $A$ is Artin's constant. Hence, for the majority of discriminants of the form $d=5p^2$, where $\bigl(\frac 5p\bigr)=1$, the class numbers are small. This is consistent with the Gauss conjecture concerning the behavior of $h(d)$ for the majority of discriminants $d>0$ in the general case.