On correlations between class numbers of imaginary quadratic fields
Volume 185 / 2018
Abstract
Let be the class number of the imaginary quadratic field with fundamental discriminant -n. We establish an asymptotic formula for correlations involving h(-n) and h(-n-l), over fundamental discriminants that avoid the congruence class 1\pmod{8}. Our result is uniform in the shift l, and the proof uses an identity of Gauss relating h(-n) to representations of integers as sums of three squares. We also prove analogous results on correlations involving r_Q(n), the number of representations of an integer n by an integral positive definite quadratic form Q.