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# The distribution of the zeros of random trigonometric polynomials

 Granville, Andrew and Wigman, Igor 2011. The distribution of the zeros of random trigonometric polynomials. American Journal of Mathematics 133 (2) , pp. 295-357. 10.1353/ajm.2011.0015

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## Abstract

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\rightarrow\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that $\frac{Z_{N}-{\Bbb E} Z_{N}}{\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.

Item Type: Article Publication Published Mathematics Q Science > QA Mathematics Johns Hopkins University Press 1080-6377 26 Jun 2019 01:57 http://orca-mwe.cf.ac.uk/id/eprint/12413