Conjecture on the moments of greatest common divisors of Gaussian ideals

Let $\mathfrak{n}$ and $\mathfrak{m}$ be nonzero ideals chosen independently and uniformly at random from the set of ideals in $\mathbb{Z}[i]$ with norm at most $x$. Write $E_x\{N(\mathfrak{n},\mathfrak{m})^n\}$ for the $n$th moment of the norm of their greatest common divisor, and let $\zeta_{\mathbb{Q}(i)}$ denote the Dedekind zeta function of $\mathbb{Q}(i)$. The moment conjecture. For $n\geq 2$,

$$E_x\{N(\mathfrak{n},\mathfrak{m})^n\}\sim \frac{4}{\pi(n+1)}\left\{\frac{2\zeta_{\mathbb{Q}(i)}(n)}{\zeta_{\mathbb{Q}(i)}(n+1)}-1\right\}x^{n-1}.$$

The preceding theorem establishes the order of growth for $n>2$ with an unspecified constant, while numerical data in the paper provide evidence for this explicit constant for all $n\geq 2$.