Small totally $p$-adic algebraic numbers
Number Theory
2019-01-11 v3
Abstract
The purpose of this note is to give a short and elementary proof of the fact, that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p-adic numbers which are neither zero nor a root of unity. The proof is based on an idea of C. Petsche and gives the best known lower bounds in this setting. These bounds differ from the truth by a term of less than .
Keywords
Cite
@article{arxiv.1802.05923,
title = {Small totally $p$-adic algebraic numbers},
author = {Lukas Pottmeyer},
journal= {arXiv preprint arXiv:1802.05923},
year = {2019}
}
Comments
There was a slight error in the citation of a result mentioned in the introduction. The strengthening of the lower bound in equation (1) does not read $\frac{1}{2}\sum_{p\in S} \frac{\log(p)}{p-1}$ but $\frac{1}{2}\sum_{p\in S} \frac{p\log(p)}{p^2-1}$. Moreover, this result is not due to P. Fili alone but due to P. Fili and C. Petsche