Sums of the floor function related to class numbers of imaginary quadratic fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915533903888384 |
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| author | Chamberland, Marc Dilcher, Karl |
| author_facet | Chamberland, Marc Dilcher, Karl |
| contents | 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 $p\equiv 1\pmod{4}$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_04387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sums of the floor function related to class numbers of imaginary quadratic fields Chamberland, Marc Dilcher, Karl Number Theory 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 $p\equiv 1\pmod{4}$. 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. |
| title | Sums of the floor function related to class numbers of imaginary quadratic fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.04387 |