Is it possible to have a continuous periodic function whose Fourier series $(a_n)_{n\in \mathbb{Z}}$ is such that $$ n . a_n \rightarrow 1?$$
I know that there exist continuous periodic functions with non-summable Fourier series, but I suspect that the sequence cannot have the proposed asymptotic behavior.
Edit. My initial question was for the condition $ |n . a_n |\rightarrow 1$, for which Conrad provides clear examples of continuous functions below, by playing on the phase of $a_n$.
Much more is true, namely, there are continuous periodic functions for which $\Sigma{|a_n|^{(2-\epsilon)}}$ diverges for any positive $\epsilon$
Construction starts by analyzing the Hardy-Littlewood series: $\Sigma e^{(icn\log n)}\frac {e^{inx}}{n^{\frac{1}{2}+\alpha}}$, where $n \geq 1$, $\alpha$ real, $c>0$ and showing for example that it converges uniformly (so to a continuos periodic function on $[0, 2\pi]$) for $0 < \alpha <1$, function which is otherwise quite complicated; in particular $\alpha = \frac{1}{2}$, gives your required example.