Linear Independence of Harmonic Numbers over the field of Algebraic Numbers

Let \(H_n =\sum\limits_{k=1}^n \frac{1}{k}\) be the \(n\)-th 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 \( 1 \leq a \leq q\) over the field of algebraic numbers. Also, for any finite set of odd primes \(J\) with \(|J|=n,\) define $$W_J=\overline{\mathbb{Q}}-\text {span of } \{ H_1, \ H_{a_{j_i}/q_i} | \ 1 \leq a_{j_i} \leq q_i -1, \ 1 \leq j_i \leq q_i-1, \ \ \forall q_i \in J\}.$$ Finally, we show that $$\text{ dim }_{\overline{\mathbb{Q}}} ~W_J=\sum\limits_{\substack{i=1 \\ q_i \in J}}^n \frac{\phi (q_i )}{2} + 2.$$