Sums of the floor function related to class numbers of imaginary quadratic fields
Marc Chamberland, Karl Dilcher
TL;DR
This work studies sums of floor-root expressions encoded in $f(n)=\sum_{j=1}^{\lfloor n/4\rfloor} \left\lfloor \sqrt{jn}\right\rfloor - \frac{n^2-1}{12}$, extending a classical identity for primes $p\equiv1\pmod{4}$ to primes $p\equiv3\pmod{4}$ and beyond. The authors connect these floor-sums to quadratic residues and, crucially, to Dirichlet's class-number formula, deriving explicit formulas for $f(p)$, $f(2p)$, and $f(4p)$ in terms of imaginary quadratic class numbers $h(-p)$ (and its variant $h^*(-p)$ for small $p$). They further determine $f(n)$ when $n$ is a prime power, with results depending on $p\bmod 4$, and propose broad conjectures for general $n$ linking $f(n)$ to $h^*(-d)$ and residual sums, revealing a deep arithmetic structure behind a simple floor-sum identity. The work thus builds a bridge between lattice-point counts and algebraic number theory, suggesting both precise formulas and rich conjectures for the interplay between floor sums and class numbers.
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 $p\equiv 1\pmod{4}$. 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.