Numerical approximation of The Masser-Gramain constant to four decimal digits: $\delta = 1.819...$
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- by Guillaume Melquiond, W. Georg Nowak and Paul Zimmermann PDF
- Math. Comp. 82 (2013), 1235-1246 Request permission
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 $k$.References
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Additional Information
- Guillaume Melquiond
- Affiliation: LRI-Bâtiment 650, Université Paris-Sud, 91405 Orsay cedex, France
- Email: guillaume.melquiond@inria.fr
- W. Georg Nowak
- Affiliation: Institute of Mathematics/DIB, Universität für Bodenkultur, Gregor Mendel-Strasse 33, 1180 Vienna, Austria
- Email: nowak@boku.ac.at
- Paul Zimmermann
- Affiliation: INRIA Nancy-Grand Est, 615 rue du Jardin Botanique, 54600 Villers-Nancy, France
- MR Author ID: 273776
- Email: Paul.Zimmermann@inria.fr
- Received by editor(s): April 8, 2011
- Received by editor(s) in revised form: September 9, 2011
- Published electronically: September 7, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 82 (2013), 1235-1246
- MSC (2010): Primary 11H06; Secondary 11P21, 52C05, 11Y60
- DOI: https://doi.org/10.1090/S0025-5718-2012-02635-4
- MathSciNet review: 3008857