Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer.
[1] The conjecture asserts that there is an absolute constant
satisfies one of the following properties: There are a number of definitions of the Mahler measure, one of which is to factor
as and then set The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial" for which the Mahler measure is the Salem number[3] It is widely believed that this example represents the true minimal value: that is,
[4][5] Consider Mahler measure for one variable and Jensen's formula shows that if
, which is also called Mahler measure.
has integer coefficients, this shows that
is an important value in the study of the integer sequences
does vanish on the circle but not at any root of unity, then the same convergence holds by Baker's theorem (in fact an earlier result of Gelfond is sufficient for this, as pointed out by Lind in connection with his study of quasihyperbolic toral automorphisms[7]).
[8] As a result, Lehmer was led to ask or Some positive answers have been provided as follows, but Lehmer's conjecture is not yet completely proved and is still a question of much interest.
be an irreducible monic polynomial of degree
Smyth[9] proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying
Blanksby and Montgomery[10] and Stewart[11] independently proved that there is an absolute constant
or[12] Dobrowolski[13] improved this to Dobrowolski obtained the value C ≥ 1/1200 and asymptotically C > 1-ε for all sufficiently large D. Voutier in 1996 obtained C ≥ 1/4 for D ≥ 2.
be an elliptic curve defined over a number field
be the canonical height function.
The canonical height is the analogue for elliptic curves of the function
The elliptic Lehmer conjecture asserts that there is a constant
If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds: due to Laurent.
[15] For arbitrary elliptic curves, the best known result is due to Masser.
[16] For elliptic curves with non-integral j-invariant, this has been improved to by Hindry and Silverman.
[17] Stronger results are known for restricted classes of polynomials or algebraic numbers.
[18] If further all the coefficients of P are odd then[19] For any algebraic number α, let
be the Mahler measure of the minimal polynomial
If the field Q(α) is a Galois extension of Q, then Lehmer's conjecture holds for
[19] The measure-theoretic entropy of an ergodic automorphism of a compact metrizable abelian group is known to be given by the logarithmic Mahler measure of a polynomial with integer coefficients if it is finite.
[20] As pointed out by Lind, this means that the set of possible values of the entropy of such actions is either all of
or a countable set depending on the solution to Lehmer's problem.
[21] Lind also showed that the infinite-dimensional torus either has ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem.
Since an ergodic compact group automorphism is measurably isomorphic to a Bernoulli shift, and the Bernoulli shifts are classified up to measurable isomorphism by their entropy by Ornstein's theorem, this means that the moduli space of all ergodic compact group automorphisms up to measurable isomorphism is either countable or uncountable depending on the solution to Lehmer's problem.