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Linear independence of harmonic numbers over the field of algebraic numbers II

  • Tapas ChatterjeeEmail author
  • Sonika Dhillon
Research Paper
  • 36 Downloads

Abstract

In a recent article, the authors have studied the arithmetic natures of a certain class of harmonic numbers. The present paper will explore the linear independence of harmonic numbers and establish the results to a more general case. More precisely, we obtain the dimension of space generated by these harmonic numbers for primes and their powers. In fact, for the case when an integer is divisible by only two distinct primes say \(m=p_1^{\alpha _1}p_2^{\alpha _2}\), define
$$\begin{aligned} W_m={\overline{{{\mathbb {Q}}}}} - \text {span of } \Big \{ H_{a/m} | \ 1 \le a \le m\Big \}. \end{aligned}$$
We prove that if m satisfies some necessary conditions, then
$$\begin{aligned} \text {dim}_{{\overline{{{\mathbb {Q}}}}}} ~W_m\le \frac{\phi (m)}{2} + 4. \end{aligned}$$
Moreover, we generalize the result to any finite set of integers of the form \(m=p_1^{\alpha _1}p_2^{\alpha _2}\) under the assumption of the same necessary conditions.

Keywords

Baker’s theory Digamma function Galois theory Gauss formula Harmonic numbers Linear forms in logarithm Primitive roots 

Mathematics Subject Classification

Primary: 11J81 11J86 15A06 Secondary: 11J91 12F10 15A03 

Notes

Acknowledgements

The authors thank Prof. M. Ram Murty and Siddhi Pathak for their comments on an earlier version of this paper. The authors would like to thank the referee for his helpful suggestions. The first author would like to thank Prof. Kohji Matsumoto and Nagoya University for the hospitality where some part of the work was done.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoparPunjabIndia

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