Self-Correlations of Hurwitz Class Numbers

The asymptotic study of class numbers of binary quadratic forms is a foundational problem in arithmetic statistics. Here, we investigate finer statistics of class numbers by studying their self-correlations under additive shifts. Specifically, we produce uniform asymptotics for the shifted convolution sum ${\sum }_{n<X}H(n)H(n+\mathrm{\ell })$ for fixed $\mathrm{\ell }\in \mathbb{Z}$, in which $H(n)$ denotes the Hurwitz class number.