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.

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.

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.