Curious Bounds for Floor Function Sums

TL;DR

This paper derives concise formulas for sums involving floor functions, providing proofs for existing bounds and proposing new conjectural bounds, advancing understanding of floor function sums.

Contribution

It introduces concise formulas for floor function sums and offers proofs for known bounds while proposing new conjectural bounds.

Findings

Provided concise formulas for floor function sums

Proved existing upper and lower bounds by Tverberg

Suggested new conjectural bounds for these sums

Abstract

The sums of floor functions have been studied by Jacobsthal, Carlitz, Grimson, and Tverberg. More recently, Onphaeng and Pongsriiam proved some sharp upper and lower bounds for the sums of Jacobsthal and Tverberg. In this paper, we devise concise formulas for the sums and then use it to give proofs of the upper and lower bounds that were claimed by Tverberg. Furthermore, we present conjectural lower and upper bounds for these sums.