For a given Dirichlet character $\chi (n) = e^{ i \theta_n}$, we prove central limit theorems for the series $\sum_{ p'} \cos \theta_{ p'}$ for non-principal characters, and $\sum_{ p' } \cos (t \log p')$ for principal characters, where $p'$ are integers based on a variant of Cram\'er's random model for the primes. For non-principal characters, we use these results to show that the Generalized Riemann Hypothesis for the associated $L$-function is true with probability equal to one. For principal characters we propose how to extend these arguments to $\Re (s) = t \to \infty$.
Categories: math.NT