Linear Independence of Harmonic Numbers over the field of Algebraic Numbers

Abstract

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

To appear in the Ramanujan Journal