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Title:
Generalised Riemann Hypothesis, Random Time Series and Normal Distributions
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Abstract:
L functions based on Dirichlet characters are natural generalizations of the Riemann zeta function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime numbers. We address the Generalized Riemann Hypothesis relative to the non-trivial complex zeros of the Dirichlet L functions by studying the possibility to enlarge the original domain of convergence of their Euler product. This leads us to analyse the asymptotic behaviour of particular series on primes: although deterministic, these series have pronounced stochastic features which make them analogous to random time series. For non-principal characters, we show that, in view of the Dirichlet theorem on the equidistribution of reduced residue classes modulo q and the Lemke Oliver-Soundararajan result on the distribution of pairs of residues on consecutive primes, these series present a universal diffusive random walk behavior with critical exponent ½.
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