# Fractional Parts and their Relations to the Values of the Riemann Zeta Function

## TL;DR

This paper establishes a new asymptotic relationship linking the sum of fractional parts multiplied by powers of integers to the values of the Riemann zeta function, generalizing classical results by Dirichlet.

## Contribution

It introduces a novel asymptotic formula connecting fractional parts and Riemann zeta function values, extending Dirichlet's classical result.

## Key findings

- Proves an asymptotic relationship involving fractional parts and zeta function values
- Generalizes Dirichlet's classical result to broader cases
- Provides mathematical insight into the connection between fractional parts and number theory

## Abstract

A well-known result, due to Dirichlet and later generalized by de la Vallee-Poussin, expresses a relationship between the sum of fractional parts and the Euler-Mascheroni constant. In this paper, we prove an asymptotic relationship between the summation of the products of fractional parts with powers of integers on one hand, and the values of the Riemann zeta function, on the other hand. Dirichlet's classical result falls as a particular case of this more general theorem.