Zeros of the dilogarithm

TL;DR

This paper investigates the zeros of the dilogarithm function, establishing their uniqueness per branch, proximity to roots of unity, and the effectiveness of Newton's method for locating them, with implications for asymptotic analysis.

Contribution

It proves that the dilogarithm has at most one zero per branch near roots of unity and demonstrates how to find these zeros precisely using Newton's method.

Findings

At most one zero per branch of the dilogarithm.

Zeros are close to roots of unity.

Newton's method efficiently finds these zeros.

Abstract

We show that the dilogarithm has at most one zero on each branch, that each zero is close to a root of unity, and that they may be found to any precision with Newton's method. This work is motivated by applications to the asymptotics of coefficients in partial fraction decompositions considered by Rademacher. We also survey what is known about zeros of polylogarithms in general.