# On correlations between class numbers of imaginary quadratic fields

## TL;DR

This paper establishes asymptotic formulas for correlations between class numbers of imaginary quadratic fields and representations by quadratic forms, using identities relating class numbers to sums of three squares.

## Contribution

It introduces uniform asymptotic formulas for correlations involving class numbers and quadratic form representations, extending previous understanding of their distribution.

## Key findings

- Asymptotic formulas for correlations involving $h(-n)$ and $h(-n-l)$.
- Results are uniform in the shift $l$.
- Analogous correlation results for representation counts $r_Q(n)$.

## Abstract

Let $h(-n)$ be the class number of the imaginary quadratic field with discriminant $-n$. We establish an asymtotic 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 $Q$.