Geodesic
Linear Independence of Logarithms of Cyclotomic Numbers and a Conjecture of Livingston
Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 31-45

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DOI

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.
DOI : 10.4153/S0008439519000468
Mots-clés : Baker’s theory, linear forms in logarithm, primitive root, units in cyclotomic field 
Chatterjee, Tapas; Dhillon, Sonika. Linear Independence of Logarithms of Cyclotomic Numbers and a Conjecture of Livingston. Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 31-45. doi: 10.4153/S0008439519000468
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     title = {Linear {Independence} of {Logarithms} of {Cyclotomic} {Numbers} and a {Conjecture} of {Livingston}},
     journal = {Canadian mathematical bulletin},
     pages = {31--45},
     year = {2020},
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