Density of gaussian primes inside consecutive disks centered along the real axis of complex plane Let's define the family of consecutive subsets of $\mathbb{N}$:
$$S_n =\{x \in \mathbb{N}\,:\,|x-n^2|\le n\}$$
With the previous definition we have that
$$U_n=\bigcup_{k=1}^n S_k=\{x \in \mathbb{N}\,:\,0\le x \le n^2+n\}$$
and
$$\pi(U_n)\sim\frac{n^2}{2\log n}$$
while
$$\pi(S_n)\sim\pi(U_n)-\pi(U_{n-1})\sim\frac{n}{\log n}$$
Therefore, the density of primes in $S_n$ is given by:
$$\rho_n=\frac{\pi(S_n)}{2n}\sim \frac{1}{2\log n}$$
Now let's extend all the previous arguments to the complex plane:
$$D_n =\{z \in \mathbb{C}\,:\,|z-n^2|\le n\}$$
$$V_n=\bigcup_{k=1}^n D_k$$

If we indicate with $\pi_G(X)$ the number of gaussian primes inside the subset $X$ of $\mathbb{C}$, numerical investigation suggests that
$$\pi_G(V_n)\sim\frac{n^3}{3\log n}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$$
while
$$\pi_G(D_n)\sim\pi_G(V_n)-\pi_G(V_{n-1})\sim\frac{n^2}{\log n}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$$
So the density of gaussian primes in $D_n$ is given by:
$$\rho_n^G\sim\frac{\pi_G(D_n)}{\pi n^2}\sim \frac{1}{\pi \log n}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)$$

I would appreciate any suggestion about theoretical validation of
asymptotic behaviors (1), (2), (3).

 A: What you observe can be explained heuristically, based on the Riemann hypothesis for the Dedekind zeta function of $\mathbb{Q}(i)$, and the expectation that $D_n$ is a not too special subregion of the annulus
$$A_n:=\{z\in\mathbb{C}:n^2-n\leq|z|\leq n^2+n\}.$$
Indeed, assuming the Riemann hypothesis for $\zeta_{\mathbb{Q}(i)}$, we get that the density of Gaussian primes in $A_n\cap\mathbb{Z}[i]$ is
$$\sim \frac{4((n^2+n)^2-(n^2-n)^2)/\log n^4}{\text{area of $A_n$}}=\frac{1}{\pi\log n}.$$
The factor $4$ is the size of the unit group $(\mathbb{Z}[i])^\times$. Perhaps this result already follows from a proven zero density theorem for $\zeta_{\mathbb{Q}(i)}$, since the analogous result for rational primes is an old result of Ingham's. At any rate, the absolute values of $z\in D_n$ vary between $n^2-n$ and $n^2+n$, and they are not too concentrated around $n^2$, so it is reasonable to expect that the density of Gaussian primes in $D_n\cap\mathbb{Z}[i]$ is asymptotically the same as in $A_n\cap\mathbb{Z}[i]$; this is what your $(3)$ records. The statements $(1)$ and $(2)$ follow readily from $(3)$. Proving $(3)$ seems nontrivial even under the Riemann hypothesis for $\zeta_{\mathbb{Q}(i)}$; but again, the known zero density theorems might be sufficient for this purpose.
