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

Research Paper

## 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

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