Computing an arithmetic constant related to the ring of Gaussian integers

Abstract:

We compute the analogue for ${\mathbf {Z}}[i]$ of Euler’s constant, that is $\delta = {\lim _{n \to + \infty }}{\delta _n}$, where ${\delta _n} = ({\Sigma _{2 \leqslant k \leqslant n}}1/\pi r_k^2) - \log n$. For this purpose we give an estimate for \[ {r_k} = \min \left \{ {r \geqslant 0;{\text {there exists}}\;z \in {\mathbf {C}}\;{\text {such that card}}({\mathbf {Z}}[i] \cap \bar D(z,r)) \geqslant k} \right \},\] and we compute a great number of values of ${r_k}$.