Expected number of real roots of random trigonometric polynomials

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials

$X_n\left(t\right)=u+\frac{1}{\sqrt{n}}\sum _{k=1}^n(A_{k}cos\left(kt\right)+B_{k}sin\left(kt\right))\text{,}t\in [0,2\pi ]\text{,}u\in \mathbb{R}$

whose coefficients $A_k,B_k$, $k\in \mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a,b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]?[0,2\pi ]$, we prove that

$\underset{n\to \infty }{lim}\frac{\mathbb{E}{N}_n[a,b]}{n}=\frac{b?a}{\pi \sqrt{3}}exp(?\frac{u^2}{2})\text{.}$