  In this paper we investigate the reproducing kernel Hilbert space where the
polylogarithm appears as kernel functions. This investigation begins with the
properties of functions in this space, and here a connection to the classical
Hardy space is shown through the Bose-Einstein integral equation. Next we
consider function theoretic operators over the polylogarithmic Hardy space,
such as multiplication and Toeplitz operators. It is determined that there are
only trivial densely defined multiplication operators (and therefore only
trivial bounded multipliers) over this space, which makes this space the first
for which this has been found to be true. In the case of Toeplitz operators, a
connection between a certain subset of these operators and the number theoretic
divisor function is found. Finally, the paper concludes with an operator
theoretic proof of the divisibility of the divisor function.
