On the geometric mean of the primitive characters

We denote by $G_n$ the geometric mean of the numbers $\phi^*(1),\phi ^*(2),\dots,\phi ^*(n)$, where $\phi ^*$ denotes the number of primitive characters modulo $q$. In this paper, we prove that \[ G_n=An +O\Big(\frac{n}{\log n}\Big), \] where $A$ is a absolute constant.