Nonreciprocal algebraic numbers of small Mahler's measure

Then M(α) ≥ 1 and, by Kronecker’s theorem, M(α) = 1 if and only if α is either zero or a root of unity. Let T ≥ 1 be a fixed real number. How many irreducible polynomials in Z[x] of degree d (or at most d) have their Mahler measures in the interval [1, T )? This question was first raised by Mignotte [12] (see also [13]) who gave the first upper bound 2(8d)2d+1 on the number of irreducible polynomials of degree d whose Mahler measures are smaller than 2. The problem was further studied in [2], [3] and [4]. In particular, an asymptotical formula for the number of integer polynomials of degree at most d and of Mahler’s measure at most T when d is fixed and T →∞ was established by Chern and Vaaler in [2]. However, the problem is much more difficult when T is small, say, fixed and d→∞. Although Kronecker’s theorem gives the answer when the interval is a singleton (the number of integer irreducible polynomials of degree at most d with Mahler’s measure 1 is equal to the number of solutions of φ(n) ≤ d, where φ is Euler’s totient function), Lehmer’s question if for each T > 1 there is an irreducible polynomial P ∈ Z[x] whose Mahler measure satisfies 1 < M(P ) < T remains open. Currently, the best upper bound for the number of irreducible polynomials of degree at most d having Mahler’s measures in [1, T ) follows from [4]. There exist at most T d+16 log d/log log d integer polynomials of degree at most d