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We prove a pointwise convergence result for additive ergodic averagesassociated with certain multiplicative actions of the Gaussian integers. Wederive several applications in dynamics and number theory, including:  (i) Wirsing's theorem for Gaussian integers: if $f\colon \mathbb{G} \to\mathbb{R}$ is a bounded completely multiplicative function, then the followinglimit exists: $$\lim_{N \to \infty} \frac{1}{N^2} \sum_{1 \leq m, n \leq N} f(m+ {\rm i} n).$$ (ii) An answer to a special case of a question ofFrantzikinakis and Host: for any completely multiplicative real-valued function$f: \mathbb{N} \to \mathbb{R}$, the following limit exists: $$\lim_{N \to\infty} \frac{1}{N^2} \sum_{1 \leq m, n \leq N} f(m^2 + n^2).$$ (iii) A variantof a theorem of Bergelson and Richter on ergodic averages along the $\Omega$function: if $(X,T)$ is a uniquely ergodic system with unique invariant measure$\mu$, then for any $x\in X$ and $f\in C(X)$,$$\lim_{N\to\infty}\frac{1}{N^2}\sum_{1 \leq m, n \leq N} f(T^{\Omega(m^2 +n^2)}x)=\int_Xf \ d\mu.$$

Averages of completely multiplicative functions over the Gaussian integers -- a dynamical approach