Improved average and almost-all bounds for $G(n)$

In this paper we study the least positive integer $G(n)$ such that the integers $a\le G(n)$ with $(a,n)=1$ generate $(\mathbb{Z}\mathrm{/}n\mathbb{Z}{)}^{\times }$. We prove that, for every $\epsilon >0$, $$\sum _{n\le x}G(n){\ll }_{\epsilon }x(\log x{)}^{8\mathrm{/}3+\epsilon },$$ improving the previously known bound $\ll x(\log x{)}^{97}$. We also prove that $G(n)\le (\log n{)}^2$ for almost all $n$, unconditionally. Thus, for almost all $n$, we obtain the same logarithmic exponent $2$ as in the classical pointwise bound under GRH.