Density of gaussian primes inside consecutive disks centered along the real axis of complex plane

Question

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).

Answer 1

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.