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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!

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up vote 6 down vote accepted

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

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Alexey, is that really the way you spell your surname? –  Robin Chapman Apr 29 '10 at 18:09
    
It's my fault: I asked Alexey (Ustinov) to answer so he didn't pay enough attention to filling in his data accurately. –  Wadim Zudilin Apr 29 '10 at 21:37
    
Now my name is correct. –  Alexey Ustinov May 4 '10 at 22:49
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