For random coefficients aj and bj we consider a random trigonometric polynomial defined as Tn(θ) = ∑n j=0{aj cos jθ + bj sin jθ}. The expected number of real zeros of Tn(θ) in the interval (0,2π) can be easily obtained. In this note we show that this number is in fact n/ √ 3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value f...

Littlewood in his 1968 monograph “Some Problems in Real and Complex Analysis” [12, problem 22] poses the following research problem, which appears to still be open: Problem. “If the nj are integral and all different, what is the lower bound on the number of real zeros of PN j=1 cos(njθ)? Possibly N −1, or not much less.” No progress appears to have been made on this in the last half century. We...

We study the number NZ(Tn) of real zeros of trigonometric polynomials

Over the past ten years, many examples of natural polynomial families from combinatorics and number theory have emerged whose zeros for high degrees appear to converge to intriguing curves in the complex plane. One interesting collection of examples appears on the website [16] of Richard Stanley which includes chromatic polynomials of complete partite graphs, q-analogue of Catalan numbers, Bern...

Using a finite element approximation, an iterative optimization scheme is described to find the z zeros of Kniz) for fixed order n. Two computer programs have been implemented to find the complex zeros with a computational accuracy of either 13 or 27 significant digits. The optimization scheme described in the paper may also be readily applied to find real and complex zeros of an arbitrary func...