Complex Zeros of Trigonometric Polynomials with Standard Normal Random Coefficients

In this paper, we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial η0 + η1 cos θ + η2 cos 2θ + · · · + ηn cos nθ in 0 2π , where the coefficients ηj = aj + ιbj , and aj n j=1 and bj n j=1 are sequences of independent normally distributed random variables with mean 0 and variance 1.W e also provide the limiting behaviour of the zeros density function as n tends to infinity.The corresponding results for the case of random algebraic polynomials are known