Lehmer’s Problem And Distance Functionals

In this thesis, we apply the operator theory and function theory to Lehmer’s problem in number theory.In Chapter One, we write Mahler’s measure as a distance functional based on Szego’s extremal theorem. Motivated by this, we introduce two operator analogues of Mahler’s measure. Later, we define a new class of operators named subharmonic operators, then the general Lehmer’s problem can be raised for each subharmonic operator. We prove that the general Lehmer’s problem fails for a class of subhar-monic operators. Some applications of these results are also given.In Chapter Two, we find that Mahler’s measure can be written as a distance functional in several function spaces. Therefore, we consider this kind of distance functional in the context of function theory. We give a necessary and sufficient condition which describes when the distance functionals map cyclotomic polynomials to 1. We prove that the general Lehmer’s conjecture holds for distance functionals on a class of reproducing function spaces on D with(?)-invariant kernels. The equivalence problems between Lehmer’s problem and the general Lehmer’s problem for ’large’ distance functionals are also revealed. An equivalent form of Lehmer’s conjecture is given by distribution function.In Chapter Three, we focus on Mahler’s measure on the unit sphere by applying techniques from function theory on polydisc, unit ball and unit sphere. The limits theorem and Kronecker type theorem are also given.