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
