Linear Independence of Logarithms of Cyclotomic Numbers and a Conjecture of Livingston

In 1965, A. Livingston conjectured the $\overline{\mathbb{Q}}$ -linear independence of logarithms of values of the sine function at rational arguments. In 2016, S. Pathak disproved the conjecture. In this article, we give a new proof of Livingston’s conjecture using some fundamental trigonometric id...

Full description

Saved in:
Bibliographic Details
Published in:Canadian mathematical bulletin 2020-03, Vol.63 (1), p.31-45
Main Authors: Chatterjee, Tapas, Dhillon, Sonika
Format: Article
Language:English
Subjects:
Algebra
Logarithms
Mathematics
Trigonometric functions
Trigonometry
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In 1965, A. Livingston conjectured the $\overline{\mathbb{Q}}$ -linear independence of logarithms of values of the sine function at rational arguments. In 2016, S. Pathak disproved the conjecture. In this article, we give a new proof of Livingston’s conjecture using some fundamental trigonometric identities. Moreover, we show that a stronger version of her theorem is true. In fact, we modify this conjecture by introducing a co-primality condition, and in that case we provide the necessary and sufficient conditions for the conjecture to be true. Finally, we identify a maximal linearly independent subset of the numbers considered in Livingston’s conjecture.
ISSN:0008-4395
1496-4287
DOI:10.4153/S0008439519000468