On the average value of a generalized Pillai function over \(\mathbb Z[i]\) in the arithmetic progression (Q395414)

Let \( \mathbb{Z}[i]\) be the ring of Gaussian integers, \(\alpha\), \(\beta \in \mathbb{Z}[i]\), \(N(\alpha)\) be the norm of \(\alpha\), and \((\alpha,\beta)\) be the greatest common divisor of \(\alpha\) and \(\beta\). For \(\alpha \in \mathbb{Z}[i]\) define the generalized Pillai function \[ g(\alpha)=\sum\limits_{\beta(\bmod\alpha)}N((\alpha,\beta)), \] where the summation is taken over all nonassociative \(\beta\) modulo \(\alpha\). The authors establish an asymptotic formula for the sum \[ \sum\limits_{\alpha=\alpha_0(\bmod\gamma)\atop N(\alpha)\leq x}g(\alpha), \] where \(\alpha_0,\gamma\in\mathbb{Z}[i]\), \((\alpha_0,\gamma)=1\). The obtained formula remains non trivial for all \(\gamma\) under the restriction \(N(\gamma)=o(x^{1/2+\varepsilon})\), \(\varepsilon>0\).