Central Limit Theorems for series of Dirichlet characters

TL;DR

This paper establishes central limit theorems for series involving Dirichlet characters, linking probabilistic behavior of these series to the truth of the Generalized Riemann Hypothesis for associated L-functions.

Contribution

It proves new central limit theorems for Dirichlet character series and connects these results to probabilistic evidence for the Generalized Riemann Hypothesis.

Findings

Central limit theorems for non-principal Dirichlet character series

Probabilistic validation of the Generalized Riemann Hypothesis

Extension proposals for principal characters as t approaches infinity

Abstract

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 $\Re (s) = t \to \infty$.