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We investigate the mean number of real zeros over an interval $[a,b]$ of arandom 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 mildassumptions on the law of the entries, we prove that this mean number isasymptotically equivalent to $\frac{n(b-a)}{\pi\sqrt{3}}$ as $n$ goes toinfinity, as in the known case of standard Gaussian coefficients. Our principalrequirement is a new Cramer type condition on the characteristic function ofthe entries which does not only hold for all continuous distributions but alsofor discrete ones in a generic sense. To our knowledge, this constitutes thefirst universality result concerning the mean number of zeros of randomtrigonometric polynomials. Besides, this is also the first time that one makesuse of the celebrated Kac-Rice formula not only for continuous random variablesas it was the case so far, but also for discrete ones. Beyond the proof of anon asymptotic version of Kac-Rice formula, our strategy consists in usingsuitable small ball estimates and Edgeworth expansions for the Kolmogorovmetric under our new weak Cramer condition, which both constitute importantbyproducts of our approach.

Universality of the mean number of real zeros of random trigonometric polynomials under a weak Cramer condition