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Linear independence of certain Lambert series

Authors: Florian Luca and Yohei Tachiya
Journal: Proc. Amer. Math. Soc. 142 (2014), 3411-3419
MSC (2010): Primary 11J72
Published electronically: June 25, 2014
MathSciNet review: 3238417
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Abstract: We prove that if $ q\ne 0,\pm 1$ and $ \ell \ge 1$ are fixed integers, then the numbers

$\displaystyle 1,\quad \sum _{n= 1} \frac {1}{q^{n}-1},\quad \sum _{n=1}^{\infty... ...{q^{n^2}-1},\quad \dots ,\quad \sum _{n=1}^{\infty }\frac {1}{q^{n^{\ell }}-1} $

are linearly independent over $ {\mathbb{Q}}$. This generalizes a result of Erdős, who treated the case $ \ell =1$. The method is based on the original approaches of Chowla and Erdős, together with some results about primes in arithmetic progressions with large moduli of Ahlford, Granville and Pomerance.

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Additional Information

Florian Luca
Affiliation: Centro de Ciencias Matemáticas, Universidad Nacional Autonoma de México, C.P. 58089, Morelia, Michoacán, México
Address at time of publication: Mathematical Institute, UNAM Juriquilla, 76230 Santiago de Querétaro, Mexico – and – School of Mathematics, University of the Witwatersrand, P. O. Box Wits 2050, South Africa

Yohei Tachiya
Affiliation: Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan

Received by editor(s): August 16, 2012
Received by editor(s) in revised form: October 31, 2012
Published electronically: June 25, 2014
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society