The Sums of the $k-$powers of the Euler set and their connection with Artin's conjecture for primitive roots

TL;DR

This paper investigates sums of powers of integers coprime to n, derives a new formula for the sum of cubes, and explores its connection to Artin's conjecture on primitive roots, linking number theory concepts.

Contribution

It introduces a new formula for the sum of cubes of Euler set elements and establishes a connection between these sums and Artin's conjecture for primitive roots.

Findings

Derived a new formula for S(3, n)

Connected S(3, p) with Artin's conjectural constant

Linked sums of powers to prime-related functions

Abstract

We examine the sums $S(k,\,n)$ of the $k-$th powers of the $\phi(n)$ integers $\alpha_1<\alpha_2<\cdots<\alpha_{\phi(n)}$ less than and prime to $n$ (Euler set) and prove a formula (new) for $S(3,\,n)$. If $n$ equals a prime $p$, we prove a theorem showing a connection of $S(3,\,p)$ with Artin's conjectural constant for primitive roots and with other functions involving primes.