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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 $n$ (Eulerset) and prove a formula (new) for $S(3,\,n)$. If $n$ equals a prime $p$, weprove a theorem showing a connection of $S(3,\,p)$ with Artin's conjecturalconstant for primitive roots and with other functions involving primes.

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