Algebraic Numbers Close to 1 and Variants of Mahler's Measure

Abstract

Given a rational functionRand a real numberp⩾1, we definehp(R) as theLpnorm of max{log|R|, 0} on the unit circle. In this paper we study the behaviour ofhp(R) providing various bounds for it. Our results lead to an explicit construction of algebraic numbers close to 1 having small Mahler's measure and small degree, which shows that a lower bound for the distance |α−1| recently given by M Mignotte and M. Waldschmidt is also sharp. From our bounds also follows a statement on polynomials equivalent to the Riemann hypothesis.