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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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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