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Universal Radial Laws for Random Polynomials: Erdős Problem #522 and Non-Universal Fluctuations
We resolve Erdős Problem #522, posed in 1961, by proving that for random Littlewood polynomials the proportion of zeros in the closed unit disk converges almost surely to $1/2$. More generally, for partial sums of random power series with i.i.d. coefficients that are bounded, nondegenerate, and cent
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