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 $O(\varphi(l))$. 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: It is easy to explain "how the Fourier analysis meshed in". Namely, using the standard notation for the Möbius function, the Euler's totient function, and the integer / fractional part functions, your sum can be written as
  $$ \sum_{n\le x} \sum_{d\mid(n,l)} \mu(d) = \sum_{d\mid l} \mu(d) \lfloor x/d \rfloor
        = x \sum_{d\mid l} \frac{\mu(d)}d + R = \frac{\phi(l)}lx + R, $$
where 
  $$ R = \sum_{d\mid l} \mu(d) \{x/d\}. $$
As Fedor Petrov observed, this already suffices to improve the remainder term from $\phi(l)$ to $\tau(l)$ and indeed, to the number of square-free divisors of $l$, which is $2^{\omega(l)}$. To get better estimates, one can try to plug in the Fourier expansion for $\{x/d\}$ and estimate the resulting sums.

As to the paper you mention, I think I was able to spot it out: is it "Extremal values of $\Delta(x,N)=\sum_{n<xN,(n,N)=1} 1-x\phi(N)$" by P. Codeca and M. Nair, published in Canad. Math. Bull. 41 (3) (1998), pp. 335–347? Another paper by the same authors on the same subject: "Links between $\Delta(x,N)=\sum_{n<xN,(n,N)=1} 1-x\phi(N)$ and character sums", Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 6 (2) (2003), pp. 509–516. I could find one more paper on this problem published in a Canadian journal: "The distribution of totatives" by D.H. Lehmer,
Canad. J. Math. 7 (1955), pp. 347–357.
A: As written above by Seva, one is led with exponential sums of the form
$$ \sum_{d|\ell} \mu(d) e^{\frac{2 i \pi y}{d}} $$
where $y = hx$ is an integer multiple of $x$. If one wants to reduce the trivial error term $O( \tau(\ell))$ to $O(\varepsilon \tau(\ell))$, one must consider the range $h \ll \frac {1}{\varepsilon}$ (at least). But I doubt that something really useful can be said about this particular sum, due to presence of the arithmetic factor $1_{d| \ell}$ (let alone the Möbius function).
If the condition $d|\ell$ is dropped (the sum is over a whole interval), then the best known results (to my knowledge) on this kind of sums are contained in this paper of Y.-F.S. Pétermann.
Note also that sieve methods give nontrivial bounds on the quantity $\sum_{n \leq x, (n, \ell) = 1} 1$ (without  Fourier analysis).
