We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $$ X_n(t)=u+\frac{ 1}{ \sqrt{ n}}\sum_{ k=1}^n (A_k\cos(kt)+B_k\sin(kt)), \quad t\in [0,2\pi],\quad u\in\mathbb{ R} $$ 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 $[a,b]\subseteq [0,2\pi]$, we prove that $$ \lim_{ n\rightarrow\infty} \frac{ \mathbb{ E} N_n[a,b]}{ n}=\frac{ b-a}{ \pi\sqrt{ 3}} e^{ -\frac{ u^2}{ 2}}. $$
Categories: math.PR