The Newman phenomenon and Lucas sequence

TL;DR

This paper provides an alternative proof of a number theoretic property involving roots of unity and extends it to relate to Lucas numbers, exploring a generalization of the Newman phenomenon.

Contribution

It introduces a new proof of a known algebraic number theory result and connects it to Lucas numbers and the Newman phenomenon.

Findings

Proves N_{Q(zeta)/Q}(1-zeta)=p for primitive p-th roots of unity.

Establishes that N_{Q(zeta)/Q}(1+zeta-zeta^2)=L(p), linking to Lucas numbers.

Explores a relation between this result and a generalization of the Newman phenomenon.

Abstract

This article gives an alternative proof of the fact that N_{Q(zeta)/Q}(1-zeta)=p where p is an odd prime number and zeta is a primitive p-th root of unity, and uses it to prove that N_{Q(zeta)/Q}(1+zeta-zeta^2)=L(p) the p-th Lucas number. It shows a relation between this result and a generalisation of the Newman phenomenon.