Linear independence of harmonic numbers over the field of algebraic numbers

Abstract

Let H n = n k=1 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 ≤ a ≤ q over the field of algebraic numbers. Also, for any finite set of odd primes J with |J| = n, defineFinally, we show that

A note on a variant of a conjecture of Rohrlich

“…The following is the lemma: Lemma For any divisor b of $$q>1$$ and $$x=1, 2, \ldots , q-1$$, $$\begin{equation} \log \sin {\left(\frac{bx \pi }{q}\right)}= (b-1) \log 2 \ + \sum \limits _{\substack{u=1 \\ u \equiv x \text{ (mod } q/b)}}^{q-1} \log \sin {\left(\frac{u \pi }{q}\right)}. \end{equation}$$For a proof, see [2, Lemma 3.9] and [4, Remark 3.4]).…”