Numerical approximation of The Masser-Gramain constant to four decimal digits: $\delta = 1.819...$

Abstract:

We prove that the constant $\delta$ studied by Masser, Gramain, and Weber, satisfies $1.819776 < \delta < 1.819833$, and disprove a conjecture of Gramain. This constant is a two-dimensional analogue of the Euler-Mascheroni constant; it is obtained by computing the radius $r_k$ of the smallest disk of the plane containing $k$ Gaussian integers. While we have used the original algorithm for smaller values of $k$, the bounds above come from methods we developed to obtain guaranteed enclosures for larger values of $k$.