On the spectrum of the Zhang-Zagier height

From recent work of Zhang and of Zagier, we know that their height $\mathfrak{H}(\alpha)$ is bounded away from 1 for every algebraic number $\alpha$ different from $0,1,1/2\pm\sqrt{-3}/2$. The study of the related spectrum is especially interesting, for it is linked to Lehmer's problem and to a conjecture of Bogomolov. After recalling some definitions, we show an improvement of the so-called Zhang-Zagier inequality. To achieve this, we need some algebraic numbers of small height. So, in the third section, we describe an algorithm able to find them, and we give an algebraic number with height $1.2875274\ldots$ discovered in this way. This search up to degree 64 suggests that the spectrum of $\mathfrak{H}(\alpha)$ may have a limit point less than 1.292. We prove this fact in the fourth part.