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

We examine the sums $S(k,n)$ of the $k-$th powers of the $\varphi (n)$ integers ${\alpha }_1<{\alpha }_2<\cdots <{\alpha }_{\varphi (n)}$ 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.