Linear independence of harmonic numbers over the field of algebraic numbers

Abstract

Let Hn = n k=1 1 k be the nth harmonic number. Euler extended it to complex arguments and defined Hr for any complex number r except for the negative integers. In this paper, we give a new proof of the transcendental nature of Hr for rational r. For some special values of q > 1, we give an upper bound for the number of linearly independent harmonic numbers Ha/q with 1 ≤ a ≤ q over the field of algebraic numbers. Also, for any finite set of odd primes J with |J | = n, define WJ = Q − span of H1, Ha j i /qi |1 ≤ a ji ≤ qi − 1, 1 ≤ ji ≤ qi − 1, ∀qi ∈ J

. Finally, we show that dim Q WJ = n i=1 qi∈J φ(qi) 2