Mean number of real zeros of a random trigonometric polynomial. III (Q1897840)

Summary: [For parts I and II, by the first author, see Proc. Am. Math. Soc. 111, No. 3, 851-863 (1991; Zbl 0722.60047) and in: Topics in polynomials of one and several variables (ed. by Th. M. Rassias et al.), 581-594 (1993).] If \(a_1, a_2, \ldots, a_n\) are independent, normally distributed random variables with mean 0 and variance 1, and if \(\nu_n\) is the mean number of zeros on the interval \((0,2 \pi)\) of the trigonometric polynomial \(a_1 \cos x + 2^{1/2} a_2 \cos 2x + \ldots+n^{1/2} a_n \cos nx\), then \[ \nu_n = 2^{- 1/2} \bigl\{ (2n+1) D_1 + (2n + 1)^{-1} D_2 + (2n + 1)^{-2} D_3 \bigr\} + O \bigl\{ (2n + 1)^{-3} \bigr\}, \] in which \(D_1 = - 0.3781 24\), \(D_2 = - 1/2\), \(D_3 = 0.5523\). After tabulation of \(5D\) values of \(\nu_n\) when \(n = 1(1) 40\), we find that the approximate formula for \(\nu_n\), obtained from the above result when the error term is neglected, produces \(5D\) values that are in error by at most \(10^{-5}\) when \(n \geq 8\), and by only about \(0.1 \%\) when \(n = 2\).