Let $0 < a < 1$ be fixed, and integer $n$ tends to infinity. It is not hard to show that the number of integers $k$ coprime to $n$ such that $1\leq k\leq an$ asymtotically equals $(a+o(1))\varphi(n)$. The question is: what are the best known estimates for the remainder and where are they written?
Many thanks!
Vinogradov, I. M. An introduction to the theory of numbers, Ch. 2, problem N 19. It gives error term $O(\tau(n))$. But direct application of inclusion-exclusion principle gives $O(2^{\omega(n)})$ (where $\tau$ is the number of divisors, and $\omega$ is the number of prime divisors. Standart solution with Mobius function also gives $O(2^{\omega(n)})$ if one take into account that $\mu(p^2)=0.$
In a similar question Bound the error in estimating a relative totient function , Alan Haynes notes in an answer that Vijayraghavan in 1951 had published a result showing many $n$ for which (for certain a) the error approaches $2^{\omega(n) - 1}$. Further, Lehmer in 1955 showed that for certain values of a (namely q/k where k divides p-1 and p divides n) that the error was 0.
