Numerical Approximation of the Masser-Gramain Constant to Four Decimal Digits: delta=1.819...

Guillaume Melquiond (1) , Werner Georg Nowak (2) , Paul Zimmermann (3)

1 TOCCATA - Formally Verified Programs, Certified Tools and Numerical Computations
2 BOKU - Universität für Bodenkultur Wien = University of Natural Resources and Life [Vienne, Autriche]
3 CARAMEL - Cryptology, Arithmetic: Hardware and Software

We prove that the constant 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.

Domaines: Algorithme et structure de données [cs.DS]

Guillaume Melquiond, Werner Georg Nowak, Paul Zimmermann. Numerical Approximation of the Masser-Gramain Constant to Four Decimal Digits: delta=1.819.... Mathematics of Computation, 2013, 82, pp.1235-1246. ⟨10.1090/S0025-5718-2012-02635-4⟩. ⟨hal-00644166⟩