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

Authors: Guillaume Melquiond, W. Georg Nowak and Paul Zimmermann
Journal: Math. Comp. 82 (2013), 1235-1246
MSC (2010): Primary 11H06; Secondary 11P21, 52C05, 11Y60
Published electronically: September 7, 2012
MathSciNet review: 3008857
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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$.

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Guillaume Melquiond
Affiliation: LRI-Bâtiment 650, Université Paris-Sud, 91405 Orsay cedex, France

W. Georg Nowak
Affiliation: Institute of Mathematics/DIB, Universität für Bodenkultur, Gregor Mendel-Strasse 33, 1180 Vienna, Austria

Paul Zimmermann
Affiliation: INRIA Nancy-Grand Est, 615 rue du Jardin Botanique, 54600 Villers-Nancy, France

Received by editor(s): April 8, 2011
Received by editor(s) in revised form: September 9, 2011
Published electronically: September 7, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.