A bound for the least Gaussian prime \(\omega\) with \(\alpha<\arg(\omega)<\beta\) (Q1581197)

Let \(\alpha, \beta\) be real numbers satisfying \(\alpha < \beta \leq \alpha + \frac{\pi}{2}\) . The author proves that there exists a Gaussian prime \(\omega\) with \(\alpha < \arg (\omega) < \beta\) such that \[ \omega \overline{\omega}< \exp \left( \frac{A}{\sqrt{\beta - \alpha}} \left( \log \frac{1}{\beta - \alpha} \right)^{3/2} \right), \] where \(A\) is a positive absolute constant. Under the assumption of the Generalized Riemann Hypothesis this estimate can be improved to \[ \omega \overline{\omega}< \frac{A}{(\beta - \alpha)^2} \log^4 \frac{1}{\beta - \alpha} . \]