Linear Independence of Logarithms of Cyclotomic Numbers and a Conjecture of Livingston. Issue 1 (12th March 2020)

• Record Type: Journal Article
• Authors: Chatterjee, Tapas; Dhillon, Sonika
• Is Part Of: Canadian mathematical bulletin =. Volume 63:Issue 1(2020)
• Page Start: 31
• Page End: 45
• Publication Date: 2020-03-12
• Subjects: 11J81, -- 11J72, -- 11J86, -- 11J91, -- 11R27; Baker's theory, -- linear forms in logarithm, -- primitive root, -- units in cyclotomic field; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439519000468
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: 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.