  We present a new method for proving non-holonomicity of sequences, which is
based on results about the number of zeros of elementary and of analytic
functions. Our approach is applicable to sequences that are defined as the
values of an elementary function at positive integral arguments. We generalize
several recent results; e.g., non-holonomicity of the logarithmic sequence is
extended to rational functions involving $\log n$. Moreover, we show that the
sequence that arises from evaluating the Riemann zeta function at odd integers
is not holonomic.
