Adamic & Huberman (2002) shows that Zipf's law is equivalent to the power law, which describes Zipf's law in a unranked form: where X is the random variable denoting the word frequency and f X (x) is its probability density function.
approximated by integration, the expected number of values in the bin [a, b] can be calculated as each bin has equal number of values within it if and only if b/a is a constant, which is in a log scale.