Fomenko, O. M. Class numbers of indefinite binary quadratic forms. (English. Russian original) Zbl 1130.11316 J. Math. Sci., New York 118, No. 1, 4918-4932 (2003); translation from Zap. Nauchn. Semin. POMI 276, 312-333 (2001). Summary: Let \(h(d)\) be the class number of properly equivalent primitive binary quadratic forms \(ax^2 + bxy + cy^2\) of discriminant \(d = b^2 a - 4ac\). 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\[ \# \left\{ {p\leq x \mid \left( {\frac{5}{p}} \right) = 1,\;h(5p^2 ) >(\log p)^{\alpha (p)} } \right\} = o(\pi (x)), \]where \[ \pi (x) = \# \{ p \leqslant x\} \] . 2. Let \(F\) be an arbitrary sufficiently large positive constant. Then for \(x >x_F\), the relation\[ \# \left\{ {p\leq x \mid \left( {\frac{5}{p}} \right) = 1,\;h(5p^2 ) >F} \right\} \asymp \frac{{\pi (x)}}{F} \]holds. 3. The relation\[ \# \left\{ {p\leq x \mid \left( {\frac{5}{p}} \right) = 1,\;h(5p^2 ) = 2} \right\} \sim \frac{9}{{19}}A\pi (x) \]holds, where \(A\) is Artin’s constant. Hence, for the majority of discriminants of the form \(d = 5p^2\), where \({\left( {\frac{5}{p}} \right) = 1}\), 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. Cited in 1 Document MSC: 11E41 Class numbers of quadratic and Hermitian forms × Cite Format Result Cite Review PDF Full Text: DOI