### 2011 Publications

#### The distribution of the zeros of random trigonometric polynomials (with Igor Wigman) American Journal of Mathematics, 133 (2011), 295-357.

We prove that the number of zeros of random trigonometric polynomials of degree $$N$$ are normally distributed with mean $$(2/\sqrt{3})N$$ and variance $$cN$$, for some constant $$c>0$$. An analogous result holds in short intervals.

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#### Prime factors of dynamical sequences (with Xander Faber) Crelle's Journal 661 (2011), 189-214.

Let $$\phi(t)\in \mathbb Q(t)$$ be of degree $$d\geq 2$$. For a given rational number $$x_0$$ , define $$x_{n+1} = \phi(x_n)$$ for each $$n\geq 0$$. If this sequence is not eventually periodic, and if $$\phi$$ does not lie in one of two explicitly determined affine conjugacy classes of rational functions, then the numerator of $$x_{n+1}-x_n$$ has a primitive prime factor for all sufficiently large $$n$$. The same result holds for the exceptional maps provided that one looks for primitive prime factors in the denominator of $$x_{n+1}-x_n$$ . Hence the result for each rational function $$\phi$$ implies (a new proof) that there are infinitely many primes.

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