| Summary: | This thesis determines some of the implications of non-universal and emergent universal statistics on arithmetic correlations and fluctuations of arithmetic functions, in particular correlations amongst prime numbers and the variance of the expected number of prime numbers over short intervals are generalised by associating these concepts to L-functions arising from number theoretic objects. Inspired by work in quantum chaology, which shares the property of displaying emergent universality, in chapter 2 a heuristic is given to determine the behaviour of a correlation function associated to functions in the Selberg class from the universal form of the 2-point correlation statistic conjectured for this class. Also in this chapter, the Riemann zeta function is taken as an example of an L-function from which the correlations between pairs of prime numbers arise from a non-universal form of the 2-point correlation statistic for its zeros. Chapter 3 explores the implications of the 2-point correlation statistic on an arithmetic variance associated to functions in the Selberg class, generalising the variance of primes in short intervals. Many of the ideas in this thesis are based on the preprint, [BKS15], joint with Hung Bui and Jon Keating.
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