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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

On the determinant of an integral lattice generated by rational approximants of the Euler constant


Authors: A. I. Aptekarev and D. N. Tulyakov
Translated by: Alex Martsinkovsky
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 70 (2009).
Journal: Trans. Moscow Math. Soc. 2009, 237-249
MSC (2000): Primary 11J72; Secondary 33C45, 41A21
Posted: December 3, 2009
MathSciNet review: 2573642
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Abstract: We investigate rational approximants of the Euler constant constructed using a certain system of jointly orthogonal polynomials and ``averaging'' such approximants of mediant type. The properties of such approximants are related to the properties of an integral lattice in  $ \mathbb{R}^3$ constructed from recurrently generated sequences. We also obtain estimates on the metric properties of a reduced basis, which imply that the $ \gamma$-forms with coefficients constructed from basis vectors of the lattice tend to zero. The question of whether the Euler constant is irrational is reduced to a property of the bases of lattices.


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Additional Information

A. I. Aptekarev
Affiliation: M. V. Keldysh Institute of Applied Mathematics, 4 Miusskaya Ploshchad′, Moscow 125047, Russia, and Lomonosov State University, Moscow, Russia
Email: aptekaa@keldysh.ru

D. N. Tulyakov
Affiliation: M. V. Keldysh Institute of Applied Mathematics, 4 Miusskaya Ploshchad′, Moscow 125047, Russia

DOI: http://dx.doi.org/10.1090/S0077-1554-09-00175-7
PII: S 0077-1554(09)00175-7
Posted: December 3, 2009
Additional Notes: Partially supported by RFFI, Project No. 08–01–00179, Program No. 1 OMN RAN, and the Support Program for Leading Scientific Schools (Project NSh–3906.2008.1).
Article copyright: © Copyright 2009 American Mathematical Society