Some sums involving Farey fractions II

Abstract

Let $F_x$ denote the Farey series of order $\left[x\right]$, i.e. the increasing sequence of irreducible fractions $p_v\in (0,1]$ whose denominators do not exceed $x$. We shall obtain precise asymptotic formulae for the sum ${{\displaystyle \sum }}_{v=1}^{\Phi \left(x\right)}{p}_v^Z$ for complex $z$ and related sums, $\Phi \left(x\right)=♯F_x$ coinciding the summatory function of Euler's function. In particular, we shall prove an asymptotic formula for ${\displaystyle \sum }{p}_v^{-1}$ with as good an estimate as for the prime number theorem by extracting an intermediate error term occurring in the asymptotic formula for $\Phi \left(x\right)$.