Sums of the floor function related to class numbers of imaginary quadratic fields

Abstract

A curious identity of Bunyakovsky (1882), made more widely known by Pólya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv 1(\mathrm{mod}4)$. We evaluate this sum also in the case $p\equiv 3(\mathrm{mod}4)$, obtaining an identity in terms of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$. We also consider certain cases where the prime $p$ is replaced by a composite integer. Class numbers of imaginary quadratic fields are again involved in some cases.