Expected number of real roots of random trigonometric polynomials

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials Xn(t)=u+1n∑k=1n(Akcos(kt)+Bksin(kt)),t∈[0,2π],u∈R whose coefficients Ak,Bk, k∈N, are independent identically distributed random variables with zero mean and unit variance. If Nn[a,b] denotes the number of real roots of Xn in an interval [a,b]⊆[0,2π], we prove that limn→∞ENn[a,b]n=b−aπ3exp(−u22).