Computing an arithmetic constant related to the ring of Gaussian integers

Abstract

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