Abstract: We consider the trigonometric polynomials $V_n(\theta)=a_0+a_1 \cos(\theta)+ \cdots +a_n \cos(n \theta)$ where $a_{2j}$ are independent identically distributed normal random variables, while $a_{2j+1}=a_{2j}$ for all $j=0, \cdots, N=(n-1)/2$ and odd $n$. We prove that for sufficient large $n$ the expected number of real zeros of $V_n$ on $[0,2\pi)$ is asymptotic to $2n/\sqrt{3}$.
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