Some sums involving Farey fractions II

To the memory of Professor Pan Cheng Dong

抄録

Let F_x denote the Farey series of order [x], i.e. the increasing sequence of irreducible fractions ρ_v∈(0, 1] whose denominators do not exceed x. We shall obtain precise asymptotic formulae for the sum \displaystyle ∑_{v=1}^¶hi(x)ρ_v^Z for complex z and related sums, ¶hi(x)=\# F_x coinciding the summatory function of Euler's function. In particular, we shall prove an asymptotic formula for \displaystyle ∑ρ_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 ¶hi(x).