On the distribution of $\hat{I}\pm p$ modulo one in imaginary quadratic number fields with class number one

Abstract

We investigate the distribution of $\hat{I}\pm p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset \mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O}=\mathbb{Z}[\ddot{I}‰]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\Vert \hat{I}\pm p{\Vert }_{\ddot{I}}‰<\mathrm{N}(p{)}^{-1/8+\hat{I}\mu }$ is satisfied for infinitely many $p$, where $\Vert \varrho {\Vert }_{\ddot{I}}‰$ measures the distance of $\varrho \in \mathbb{C}$ to $\mathcal{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

36 pages