# On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one

## TL;DR

This paper studies how the products of a fixed complex number and prime elements in imaginary quadratic fields distribute modulo one, extending classical results using advanced sieve methods and harmonic analysis techniques.

## Contribution

It extends distribution results of $eta p$ modulo one to imaginary quadratic fields with class number one, employing Harman's sieve and Poisson summation in a novel setting.

## Key findings

- Infinitely many primes p satisfy the inequality involving $
orm{eta p}_ ext{omega}$
- Established distribution bounds analogous to classical results in new number field context
- Introduced smoothing techniques for effective use of Poisson summation in algebraic number fields

## Abstract

We investigate the distribution of $\alpha p$ 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 $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega < \mathrm{N}(p)^{-1/8+\epsilon}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_\omega$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ 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.