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

2025/10/05 by Marc Chamberland, Karl Dilcher, Chamberland, Marc +1 · 1 citation
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2510.04387

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≡ 1\pmod4. We evaluate this sum also in the case p≡ 3\pmod4, obtaining an identity in terms of the class number of the imaginary quadratic field \mathbb Q(√(-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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