# Sums of certain fractional parts

## TL;DR

This paper establishes an improved upper bound for sums of fractional parts of smooth functions, leveraging Weyl's bound and Popov's technique, with applications in analytic number theory.

## Contribution

It introduces a novel upper bound for fractional part sums by combining Weyl's bound and Popov's method, advancing analytic number theory techniques.

## Key findings

- Improved upper bound for fractional part sums
- Enhanced main term estimation using Weyl's bound
- Application potential in analytic number theory

## Abstract

In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl's bound for exponential sums and a device used by Popov.