Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected number of roots in various sets. This is done for polynomials with coefficients that may be dependent, and need not have identical distributions. We also study random polynomials spanned by various deterministic bases.

Let α, β ∈ R be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence B α,β = (αn + β) ∞ n=1. We conjecture that the same result holds true when α is an irrational number of infinite type.

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Let {ϕ i } ∞ i=0 be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure µ. We study zero distribution of random linear combinations of the formwhere η 0 , . . . , η n−1 are i.i.d. standard Gaussian variables. We use the Christoffel-Darboux formula to simplify the density functions provided by Vanderbei for the expected number real and complex of zeros of Pn. From these expressions, under the assumption that µ is in the Nevai class, we deduce the limiting value of these density functions away from the unit circle. Under the mere assumption that µ is doubling on subarcs of T centered at 1 and −1, we show that the expected number of real zeros of Pn is at most (2/π) log n + O(1), and that the asymptotic equality holds when the corresponding recurrence coefficients decay no slower than n −(3+ǫ)/2 , ǫ > 0. We conclude with providing results that estimate the expected number of complex zeros of Pn in shrinking neighborhoods of compact subsets of T.

Let {f j } n j=0 be a sequence of orthonormal polynomials where the orthogonality relation is satisfied on either the real line or on the unit circle. We study zero distribution of random linear combinations of the formwhere η 0 , . . . , η n are complex-valued i.i.d. standard Gaussian random variables. Using the Christoffel-Darboux formula, the density function for the expected number of zeros of P n in these cases takes a very simple shape. From these expressions, under the mere assumption that the orthogonal polynomials are from the Nevai class, we give the limiting value of the density function away from their respective sets where the orthogonality holds. In the case when {f j } are orthogonal polynomials on the unit circle, the density function shows that the expected number of zeros of P n are clustering near the unit circle. To quantify this phenomenon, we give a result that estimates the expected number of complex zeros of P n in shrinking neighborhoods of compact subsets of the unit circle.