  Zipf's law appears in many application areas but does not have a closed form
expression, which may make its use cumbersome. Since it coincides with the
truncated version of the Zeta distribution, in this paper we propose three
approximate closed form expressions for the truncated Zeta distribution, which
may be employed for Zipf's law as well. The three approximations are based on
the replacement of the sum occurring in Zipf's law with an integral, and are
named respectively the integral approximation, the average integral
approximation, and the trapezoidal approximation. While the first one is shown
to be of little use, the trapezoidal approximation exhibits an error which is
typically lower than 1\%, but is as low as 0.1\% for the range of values of the
Zipf parameter below 1.
