On correlations between class numbers of imaginary quadratic fields
Volume 185 / 2018
Acta Arithmetica 185 (2018), 211-231
MSC: Primary 11E25; Secondary 11R29, 11P55.
DOI: 10.4064/aa170319-13-12
Published online: 6 July 2018
Abstract
Let $h(-n)$ 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 $Q$.
