Complex zeros of trigonometric polynomials with standard normal random coefficients (Q5953086)

Consider the random trigonometric polynomial NEWLINE\[NEWLINET_n(\vartheta):= \sum^n_{j= 0} \eta_j\cos j\vartheta,NEWLINE\]NEWLINE where the coefficients \(\eta_j= a_j+ ib_j\), and \(\{a_j\}\), \(\{b_j\}\) are sequences of independent normally distributed random variables with mean \(0\) and variance \(1\). The authors obtain an exact formula for the average density of the distribution of the complex zeros of \(T_n(\vartheta)\). They also provide the limiting behaviour of the zeros density function as \(n\to\infty\).NEWLINENEWLINENEWLINEThe corresponding results for the case of random algebraic polynomials are known [see, for example, \textit{A. T. Bharucha-Reid} and \textit{M. Sambandham}, ``Random polynomials'' (1986; Zbl 0615.60058)].