  We prove a pointwise convergence result for additive ergodic averages
associated with certain multiplicative actions of the Gaussian integers. We
derive 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 following
limit 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 of
Frantzikinakis 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 variant
of 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 $f\in C(X)$,
$$\lim_{N\to\infty}\frac{1}{N^2}\sum_{1 \leq m, n \leq N} f(T^{\Omega(m^2 +
n^2)}x)=\int_Xf \ d\mu.$$
