Lower bound on height of algebraic numbers and low lying zeros of the Dedekind zeta-function

TL;DR

This paper links the Weil height of algebraic integers to zeros of Dedekind zeta-functions, proving Lehmer's conjecture for some infinite extensions under GRH and exploring prime ideal conditions.

Contribution

It provides new lower bounds on algebraic number heights and establishes Lehmer's conjecture for specific infinite extensions assuming GRH.

Findings

Lower bounds on Weil height in terms of zeta zeros

Proof of Lehmer's conjecture for certain infinite extensions under GRH

Introduction of prime ideal conditions for infinite extensions

Abstract

In this paper, we establish lower bounds on Weil height of algebraic integers in terms of the low lying zeros of the Dedekind zeta-function. As a result, we prove Lehmer's conjecture for certain infinite non-Galois extensions conditional on GRH. We also introduce and study a condition on prime ideals with small norms for arbitrary infinite extensions, in the spirit of a prime splitting condition for infinite Galois extensions introduced by E. Bombieri and U. Zannier.