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

Part of: Algebraic number theory: global fields Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  12 December 2019

Tapas Chatterjee  and
Sonika Dhillon
Tapas Chatterjee
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, Punjab, India Email: tapasc@iitrpr.ac.insonika@iitrpr.ac.in
Sonika Dhillon
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, Punjab, India Email: tapasc@iitrpr.ac.insonika@iitrpr.ac.in
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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.

Keywords

Baker’s theorylinear forms in logarithmprimitive rootunits in cyclotomic field

MSC classification

Primary: 11J81: Transcendence (general theory)
Secondary: 11J72: Irrationality; linear independence over a field 11J86: Linear forms in logarithms; Baker's method 11J91: Transcendence theory of other special functions 11R27: Units and factorization
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 31 - 45
Copyright
© Canadian Mathematical Society 2019

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References

Baker, A., Transcendental number theory. Cambridge University Press, London–New York, 1975.CrossRefGoogle Scholar
Bass, H., Dirichlet unit theorem, induced characters, and Whitehead groups of finite groups. Topology 4(1965), 391–410. https://doi.org/10.1016/0040-9383(66)90036-XCrossRefGoogle Scholar
Chatterjee, T. and Dhillon, S., Linear independence of harmonic numbers over the field of algebraic numbers. Ramanujan J. (2019). https://doi.org/10.1007/s11139-018-0124-6CrossRefGoogle Scholar
Chatterjee, T. and Dhillon, S., Linear independence of harmonic numbers over the field of algebraic numbers II. Ramanujan J. https://doi.org/10.1007/s11139-019-00183-8Google Scholar
Chatterjee, T. and Gun, S., The digamma function, Euler–Lehmer constants and their p-adic counterparts. Acta Arith. 162(2014), 197–208. https://doi.org/10.4064/aa162-2-4CrossRefGoogle Scholar
Chatterjee, T. and Murty, M. R., Non-vanishing of Dirichlet series with periodic coefficients. J. Number Theory 145(2014), 1–21. https://doi.org/10.1016/j.jnt.2014.05.010CrossRefGoogle Scholar
Chatterjee, T., Murty, M. R., and Pathak, S., A vanishing criterion for Dirichlet series with periodic coefficients. In: Number theory related to modular curves—Momose memorial volume. Contemp. Math., 701, American Mathematical Society, Providence, RI, 2018, pp. 69–80. https://doi.org/10.1090/conm/701/14141CrossRefGoogle Scholar
Ennola, V., On relations between cyclotomic units. J. Number Theory 4(1972), 236–247. https://doi.org/10.1016/0022-314X(72)90050-9CrossRefGoogle Scholar
Livingston, A., The series ∑n=1∞f (n)/n for periodic f. Canad. Math. Bull. 8(1965), 413–432. https://doi.org/10.4153/CMB-1965-029-2CrossRefGoogle Scholar
Murty, M. R. and Saradha, N., Transcendental values of the p-adic digamma function. Acta Arith. 133(2008), 349–362. https://doi.org/10.4064/aa133-4-4CrossRefGoogle Scholar
Murty, M. R. and Saradha, N., Euler–Lehmer constants and a conjecture of Erdős. J. Number Theory 130(2010), 2671–2682. https://doi.org/10.1016/j.jnt.2010.07.004CrossRefGoogle Scholar
Pathak, S., On a conjecture of Livingston. Canad. Math. Bull. 60(2017), 184–195. https://doi.org/10.4153/CMB-2016-065-1CrossRefGoogle Scholar
Pei, D. Y. and Feng, K., A note on the independence of the cyclotomic units. Acta Math. Sinica 23(1980), 773–778.Google Scholar
Ramachandra, K., On the units of cyclotomic fields. Acta Arith. 12(1966/67), 165–173. https://doi.org/10.4064/aa-12-2-165-173CrossRefGoogle Scholar
Washington, L. C., Introduction to cyclotomic fields, Second ed., Graduate Texts in Mathematics, 83, Springer, New York, 1997. https://doi.org/10.1007/978-1-4612-1934-7CrossRefGoogle Scholar