A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for
$$\sum_{n \leq x, (n, \ell) = 1} 1$$
Of course, this is easy to estimate with a trivial error term of $O(\varphi(q))$. O(\varphi(l))$. However in the paper I am looking for the authors attempted obtaining better bounds, using some Fourier analysis (in particular the Fourier series for the fractional part of x). I think, bounds in the sum $$\sum_{n \leq x} (n, \ell)$$ are essentially an equivalent variation of the problem, so references on this problem are welcome aswell. The reason why I am interested in this problem is ... pure curiosity. I am curious to see how the Fourier methods meshed in, and what kind of bounds they gave, even though of course we cannot really expect anything too fantastic in this problem. 1 # Number of integers coprime to l A long time ago I've seen a paper considering, given$\ell$fixed, estimates for $$\sum_{n \leq x, (n, \ell) = 1} 1$$ Of course, this is easy to estimate with a trivial error term of$O(\varphi(q))\$. However in the paper I am looking for the authors attempted obtaining better bounds, using some Fourier analysis (in particular the Fourier series for the fractional part of x). I think, bounds in the sum
$$\sum_{n \leq x} (n, \ell)$$ are essentially an equivalent variation of the problem, so references on this problem are welcome aswell.