Approximation of the truncated Zeta distribution and Zipf's law

TL;DR

This paper introduces three approximate closed-form expressions for the truncated Zeta distribution, facilitating easier use of Zipf's law in various applications by providing accurate and computationally efficient approximations.

Contribution

The paper proposes three novel approximation methods for the truncated Zeta distribution, improving the practical usability of Zipf's law with high accuracy.

Findings

Trapezoidal approximation achieves errors below 1%.

Integral approximation is less useful due to higher errors.

Approximate formulas are effective for Zipf's law applications.

Abstract

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.