Reciprocity Relations for Summations of Squares of Floor Functions and Fractional Parts of Fractions

TL;DR

This paper derives reciprocity relations that enable efficient computation of sums involving squares of fractional parts and floors of fractions for coprime integers, advancing number theory techniques.

Contribution

It introduces new reciprocity relations for summations of squared fractional parts and floors, providing a faster evaluation method for these sums.

Findings

Derived explicit reciprocity formulas for sums of fractional parts squared

Established efficient evaluation techniques for summations involving floors and fractional parts

Enhanced computational methods in number theory for coprime integers

Abstract

Given positive coprime integers $a$ and $b$ and a natural number $h$, we obtain reciprocity relations which can be used to quickly evaluate summations like $\sum_{i=1}^{h} \{\frac{ib}{a}\}^2$ and $\sum_{i=1}^{h} \lfloor \frac{ib}{a} \rfloor^2$, where $\lfloor x \rfloor$ and $\{x\}$ denote the floor function and the fractional part of $x$, respectively.