Large Moments and Extreme Values of Class Numbers of Indefinite Binary Quadratic Forms

Abstract

Let h(d) be the class number of indefinite binary quadratic forms of discriminant d, and let ε d be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the k-th moment of h(d) over positive discriminants d with ε d ≤ x, uniformly for real numbers k in the range 0 < k ≤ (log x) 1−o(1) . This improves upon the work of Raulf, who obtained such an asymptotic for a fixed positive integer k. We also investigate the distribution of large values of h(d) when the d's are ordered according to the size of their fundamental units ε d . In particular, we show that the tail of this distribution has the same shape as that of class numbers of imaginary quadratic fields ordered by the size of their discriminants. As an application of these results, we prove that there are many positive discriminants d with class number h(d) ≥ (e γ /3 + o(1)) · ε d (log log ε d )/ log ε d , a bound that we believe is best possible. We also obtain an upper bound for h(d) that is twice as large, assuming the generalized Riemann hypothesis.