The square-root cancellation conjecture for prime values of non-principal Dirichlet characters

Let $p_n$ be the $n$th prime, and let $\chi$ be a non-principal Dirichlet character modulo the positive integer $k$. For primes with $\chi(p_n)\neq 0$, write

$$\chi(p_n)=e^{i\theta_{p_n}}.$$

Define

$$C_N=\sum_{\substack{n=1\\ p_n\ne k}}^N\cos\theta_{p_n}.$$

Square-root cancellation conjecture. As $N\to\infty$,

$$C_N=O(\sqrt{N}).$$

The paper uses this conjectural bound as the main hypothesis for proving convergence of the associated Euler product in the right half of the critical strip and consequently constraining non-trivial zeros to the critical line. It is presented as unproved, with heuristic support from a random-walk model.