Hostname: page-component-7dd5485656-tbj44 Total loading time: 0 Render date: 2025-10-29T21:03:36.483Z Has data issue: false hasContentIssue false
  • English
  • Français

IRRATIONALITY AND NONQUADRATICITY MEASURES FOR LOGARITHMS OF ALGEBRAIC NUMBERS

Part of: Miscellaneous topics of analysis in the complex domain Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  22 November 2012

RAFFAELE MARCOVECCHIO  and
CARLO VIOLA
RAFFAELE MARCOVECCHIO
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria (email: marcovr8@univie.ac.at)
CARLO VIOLA*
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy (email: viola@dm.unipi.it)
*
✉For correspondence; e-mail: viola@dm.unipi.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let 𝕂⊂ℂ be a number field. We show how to compute 𝕂-irrationality measures of a number ξ∉𝕂, and 𝕂-nonquadraticity measures of ξ if [𝕂(ξ):𝕂]>2. By applying the saddle point method to a family of double complex integrals, we prove ℚ(α)-irrationality measures and ℚ(α)-nonquadraticity measures of log α for several algebraic numbers α∈ℂ, improving earlier results due to Amoroso and the second-named author.

Keywords

irrationality measurenonquadraticity measureWeil heightsaddle point method

MSC classification

Secondary: 11J82: Measures of irrationality and of transcendence 11J72: Irrationality; linear independence over a field 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 237 - 267
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[AV]Amoroso, F. and Viola, C., ‘Approximation measures for logarithms of algebraic numbers’, Ann. Sc. Norm. Super Pisa (4) 30 (2001), 225–249.Google Scholar
[H1]Hata, M., ‘Rational approximations to π and some other numbers’, Acta Arith. 63 (1993), 335–349.CrossRefGoogle Scholar
[H2]Hata, M., ‘ℂ2-saddle method and Beukers’ integral’, Trans. Amer. Math. Soc. 352 (2000), 4557–4583.Google Scholar
[M]Marcovecchio, R., ‘The Rhin–Viola method for log 2’, Acta Arith. 139 (2009), 147–184.Google Scholar
[RV1]Rhin, G. and Viola, C., ‘On a permutation group related to ζ(2)’, Acta Arith. 77 (1996), 23–56.CrossRefGoogle Scholar
[RV2]Rhin, G. and Viola, C., ‘The group structure for ζ(3)’, Acta Arith. 97 (2001), 269–293.CrossRefGoogle Scholar
[RV3]Rhin, G. and Viola, C., ‘The permutation group method for the dilogarithm’, Ann. Sc. Norm. Super Pisa (5) 4 (2005), 389–437.Google Scholar
[S]Schmidt, W. M., Diophantine Approximation, Lecture Notes in Mathematics, 785 (Springer, 1980).Google Scholar
[V]Viola, C., ‘Hypergeometric functions and irrationality measures’, in: Analytic Number Theory (Kyoto, 1996), London Mathematical Society Lecture Note Series, 247 (Cambridge University Press, Cambridge, 1997), pp. 353–360.CrossRefGoogle Scholar