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On a probability space $(\Omega, \mathcal F, \mathbb P)$ we consider twoindependent sequences $(a_k)_{k \geq 1}$ and $(b_k)_{k \geq 1}$ of i.i.d.random variables that are centered with unit variance and which admit a momentstrictly higher than two. We define the associated random trigonometricpolynomial \[ f_n(t) :=\frac{1}{\sqrt{n}} \sum_{k=1}^n a_k \cos(kt)+b_k\sin(kt), \quad t \in \mathbb R. \] In their seminal work, for Rademachercoefficients, Salem and Zygmund showed that $\mathbb P$ almost surely: \[\forall t\in\mathbb R,\quad \frac{1}{2\pi}\int_{0}^{2\pi} \exp\left(i tf_n(x)\right) dx \xrightarrow[n\to\infty]~e^{-\frac{t^2}{2}}. \] In otherwords, if $X$ denotes an independent random variable uniformly distributed on$[0,2\pi]$, $\mathbb{P}$ almost surely, under the law of $X$, $f_n(X)$converges in distribution to a standard Gaussian variable. In this paper, werevisit the above result from different perspectives. Namely, i) we establish apossibly sharp convergence rate for some adequate metric via the Stein'smethod, ii) we prove a functional counterpart of Salem--Zygmund CLT, iii) weextend it to more general distributions for $X$, iv) we also prove that theconvergence actually holds in total variation. As an application, in the casewhere the random coefficients have a symmetric distribution and admit a momentof order $4$, we show that $\mathbb{P}$ almost surely, for any interval $[a,b]\subset [0, 2\pi]$ \[\frac{\mathcal N(f_n,[a,b])}{n} \xrightarrow[n \to+\infty]{} \frac{(b-a)}{\pi \sqrt{3}},\] where $\mathcal N(f_n,[a,b])$ denotesthe number of real zeros of $f_n$ in the interval $[a,b]$. To the best of ourknowledge, such an almost sure result is new in the framework of randomtrigonometric polynomials, even in the case of Gaussian coefficients.

Variations on Salem--Zygmund results for random trigonometric polynomials. Application to almost sure nodal asymptotics