Linear Independence of Harmonic Numbers over the field of Algebraic Numbers

TL;DR

This paper investigates the linear independence of harmonic numbers over algebraic numbers, providing new proofs of their transcendence and calculating the dimension of related spans for sets of harmonic numbers with rational indices.

Contribution

It offers a new proof of the transcendental nature of harmonic numbers at rational points and determines the dimension of their spans over algebraic numbers for specific prime sets.

Findings

Proves the transcendental nature of harmonic numbers at rational points.

Provides an upper bound for the linear independence of harmonic numbers with rational indices.

Calculates the exact dimension of the span of harmonic numbers over algebraic numbers for sets of primes.

Abstract

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 $$\text{ dim }_{\overline{\mathbb{Q}}} ~W_J=\sum\limits_{\substack{i=1 \\ q_i \in J}}^n \frac{\phi (q_i )}{2} + 2.$$