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Linear independence of harmonic numbers over the field of algebraic numbers

  • Tapas ChatterjeeEmail author
  • Sonika Dhillon
Article
  • 39 Downloads

Abstract

Let \(H_n =\sum _{k=1}^n \frac{1}{k}\) be the nth harmonic number. Euler extended it to complex arguments and defined \(H_r\) for any complex number r except for the negative integers. In this paper, we give a new proof of the transcendental nature of \(H_r\) for rational r. For some special values of \(q>1,\) we give an upper bound for the number of linearly independent harmonic numbers \(H_{a/q}\) with \( 1 \le a \le q\) over the field of algebraic numbers. Also, for any finite set of odd primes J with \(|J|=n,\) define
$$\begin{aligned} W_J=\overline{{\mathbb {Q}}}-\text {span of } \big \{H_1, H_{a_{j_i}/q_i} | 1 \le a_{j_i} \le q_i -1, \; 1 \le j_i \le q_i-1, \;\forall q_i \in J\big \}. \end{aligned}$$
Finally, we show that
$$\begin{aligned} \text { dim }_{\overline{{\mathbb {Q}}}} ~W_J=\sum \limits _{\begin{array}{c} i=1 \\ q_i \in J \end{array}}^n \frac{\phi (q_i )}{2} + 2. \end{aligned}$$

Keywords

Baker’s Theory Digamma function Galois theory Gauss formula Harmonic Numbers Linear forms in logarithm Linear independence 

Mathematics Subject Classification

Primary 11J81 11J86 Secondary 11J91 

Notes

Acknowledgements

The authors thank Prof. M. Ram Murty for his comments on an earlier version of this paper. The authors would like to thank the referee for his insightful suggestions. The authors also thank Mr. Suraj Singh Khurana for his help in the revision process of this article. The first author would like to thank Université Pierre et Marie Curie (University of Paris VI) for the hospitality where some part of the work was done and the National Board for Higher Mathematics for the partial support.

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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 RoparRupnagarIndia

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