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

Guillaume Melquiond 1, 2 W. Georg Nowak 3 Paul Zimmermann 4
2 TOCCATA - Certified Programs, Certified Tools, Certified Floating-Point Computations
LRI - Laboratoire de Recherche en Informatique, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, CNRS - Centre National de la Recherche Scientifique : UMR8623
4 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : 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 rk 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.
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Submitted on : Wednesday, November 23, 2011 - 5:11:18 PM
Last modification on : Tuesday, December 18, 2018 - 4:18:25 PM

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

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