Let denote the Farey series of order , i.e. the increasing sequence of irreducible fractions whose denominators do not exceed . We shall obtain precise asymptotic formulae for the sum for complex and related sums, coinciding the summatory function of Euler's function. In particular, we shall prove an asymptotic formula for with as good an estimate as for the prime number theorem by extracting an intermediate error term occurring in the asymptotic formula for .
Shigeru KANEMITSU. Takako KUZUMAKI. Masami YOSHIMOTO. "Some sums involving Farey fractions II." J. Math. Soc. Japan 52 (4) 915 - 947, October, 2000. https://doi.org/10.2969/jmsj/05240915