  Let us consider i.i.d. random variables $\{a_k,b_k\}_{k \geq 1}$ defined on a
common probability space $(\Omega, \mathcal F, \mathbb P)$, following a
symmetric Rademacher distribution and the associated random trigonometric
polynomials $S_n(\theta)= \frac{1}{\sqrt{n}} \sum_{k=1}^n a_k \cos(k\theta)+b_k
\sin(k\theta)$. A seminal result by Salem and Zygmund ensures that
$\mathbb{P}-$almost surely, $\forall t\in\mathbb{R}$ \[ \lim_{n \to +\infty}
\frac{1}{2\pi}\int_0^{2\pi} e^{i t S_n(\theta)}d\theta=e^{-t^2/2}. \] This
result was then further generalized in various directions regarding the
coefficients distribution, their dependency structure or else the dimension and
the nature of the ambient manifold. To the best of our knowledge, the natural
question of the fluctuations in the above limit has not been tackled so far and
is precisely the object of this article. Namely, for general i.i.d. symmetric
random coefficients having a finite sixth-moment and for a large class of
continuous test functions $\phi$ we prove that \[
\sqrt{n}\left(\frac{1}{2\pi}\int_0^{2\pi}
\phi(S_n(\theta))d\theta-\int_{\mathbb{R}}\phi(t)\frac{e^{-\frac{t^2}{2}}dt}{\sqrt{2\pi}}\right)\xrightarrow[n\to\infty]{\text{Law}}~\mathcal{N}\left(0,\sigma_{\phi}^2+\frac{c_2(\phi)^2}{2}\left(\mathbb{E}(a_1^4)-3\right)\right).
\] Here, the constant $\sigma_{\phi}^2$ is explicit and corresponds to the
limit variance in the case of Gaussian coefficients and $c_2(\phi)$ is the
coefficient of order $2$ in the decomposition of $\phi$ in the Hermite
polynomial basis. Surprisingly, it thus turns out that the fluctuations are not
universal since they both involve the kurtosis of the coefficients and the
second coefficient of $\phi$ in the Hermite basis.
