  We investigate the mean number of real zeros over an interval $[a,b]$ of a
random trigonometric polynomial of the form $\sum_{k=1}^n a_k \cos(kt)+b_k
\sin(kt)$ where the coefficients are i.i.d. random variables. Under mild
assumptions on the law of the entries, we prove that this mean number is
asymptotically equivalent to $\frac{n(b-a)}{\pi\sqrt{3}}$ as $n$ goes to
infinity, as in the known case of standard Gaussian coefficients. Our principal
requirement is a new Cramer type condition on the characteristic function of
the entries which does not only hold for all continuous distributions but also
for discrete ones in a generic sense. To our knowledge, this constitutes the
first universality result concerning the mean number of zeros of random
trigonometric polynomials. Besides, this is also the first time that one makes
use of the celebrated Kac-Rice formula not only for continuous random variables
as it was the case so far, but also for discrete ones. Beyond the proof of a
non asymptotic version of Kac-Rice formula, our strategy consists in using
suitable small ball estimates and Edgeworth expansions for the Kolmogorov
metric under our new weak Cramer condition, which both constitute important
byproducts of our approach.
