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arXiv · 2510.04387

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

Marc Chamberland·Karl Dilcher
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Abstract

A curious identity of Bunyakovsky (1882), made more widely known by P\'olya 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 $p\equiv 1\pmod{4}$. We evaluate this sum also in the case $p\equiv 3\pmod{4}$, 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.

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Marc Chamberland, Karl Dilcher. 2025-10-05. Sums of the floor function related to class numbers of imaginary quadratic fields. https://arxiv.org/abs/2510.04387

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