Lehmer's conjecture

For Lehmer's conjecture about the non-vanishing of τ(n), see Ramanjuan's tau function.

For Lehmer's conjecture about Euler's totient function, see Lehmer's totient problem.

Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant $\mu>1$ such that every polynomial with integer coefficients $P(x)\in\mathbb{Z}[x]$ satisfies one of the following properties:

• The Mahler measure $\mathcal{M}(P(x))$ of $P(x)$ is greater than or equal to $\mu$.

• $P(x)$ is an integral multiple of a product of cyclotomic polynomials or the monomial $x$, in which case $\mathcal{M}(P(x))=1$. (Equivalently, every complex root of $P(x)$ is a root of unity or zero.)

There are a number of definitions of the Mahler measure, one of which is to factor $P(x)$ over $\mathbb{C}$ as

$P(x)=a_0 (x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_D),$

and then set

$\mathcal{M}(P(x)) = |a_0| \prod_{i=1}^{D} \max(1,|\alpha_i|).$

The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial"

$P(x)= x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1 \,,$

for which the Mahler measure is the Salem number

$\mathcal{M}(P(x))=1.176280818\dots \ .$

It is widely believed that this example represents the true minimal value: that is, $\mu=1.176280818\dots$ in Lehmer's conjecture.

Motivation

Consider Mahler measure for one variable and Jensen's formula shows that if $P(x)=a_0 (x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_D)$ then

$\mathcal{M}(P(x)) = |a_0| \prod_{i=1}^{D} \max(1,|\alpha_i|).$

In this paragraph denote　$m(P)=\log(\mathcal{M}(P(x))$ , which is also called Mahler measure.

If $P$ has integer coefficients, this shows that $\mathcal{M}(P)$ is an algebraic number so $m(P)$ is the logarithm of an algebraic integer. It also shows that $m(P)\ge0$ and that if $m(P)=0$ then $P$ is a product of cyclotomic polynomials i.e. monic polynomials whose all roots are roots of unity, or a monomial polynomial of $x$ i.e. a power $x^n$ for some $n$ .

Lehmer noticed that $m(P)=0$ is an important value in the study of the integer sequences $\Delta_n=\text{Res}(P(x), x^n-1)=\prod^D_{i=1}(\alpha_i^n-1)$ for monic $P$ . If $P$ does not vanish on the circle then $\lim|\Delta_n|^{1/n}=\mathcal{M}(P)$ and this statement might be true even if $P$ does vanish on the circle. By this he was led to ask

whether there is a constant $c>0$ such that $m(P)>c$ provided $P$ is not cyclotomic?,

or

given $c>0$, are there $P$ with integer coefficients for which $0<m(P)<c$?

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.

Partial results

Let $P(x)\in\mathbb{Z}[x]$ be an irreducible monic polynomial of degree $D$.

Smyth proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying $x^DP(x^{-1})\ne P(x)$.

Blanksby and Montgomery and Stewart independently proved that there is an absolute constant $C>1$ such that either $\mathcal{M}(P(x))=1$ or

$\log\mathcal{M}(P(x))\ge \frac{C}{D\log D}.$

Dobrowolski improved this to

$\log\mathcal{M}(P(x))\ge C\left(\frac{\log\log D}{\log D}\right)^3.$

Dobrowolski obtained the value C ≥ 1/1200 and asymptotically C > 1-ε for all sufficiently large D. Voutier obtained C ≥ 1/4 for D ≥ 2.

Elliptic analogues

Let $E/K$ be an elliptic curve defined over a number field $K$, and let $\hat{h}_E:E(\bar{K})\to\mathbb{R}$ be the canonical height function. The canonical height is the analogue for elliptic curves of the function $(\deg P)^{-1}\log\mathcal{M}(P(x))$. It has the property that $\hat{h}_E(Q)=0$ if and only if $Q$ is a torsion point in $E(\bar{K})$. The elliptic Lehmer conjecture asserts that there is a constant $C(E/K)>0$ such that

$\hat{h}_E(Q) \ge \frac{C(E/K)}{D}$ for all non-torsion points $Q\in E(\bar{K})$,

where $D=[K(Q):K]$. If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds:

$\hat{h}_E(Q) \ge \frac{C(E/K)}{D} \left(\frac{\log\log D}{\log D}\right)^3 ,$

due to Laurent. For arbitrary elliptic curves, the best known result is

$\hat{h}_E(Q) \ge \frac{C(E/K)}{D^3(\log D)^2},$

due to Masser. For elliptic curves with non-integral j-invariant, this has been improved to

$\hat{h}_E(Q) \ge \frac{C(E/K)}{D^2(\log D)^2},$

by Hindry and Silverman.

Restricted results

Stronger results are known for restricted classes of polynomials or algebraic numbers.

If P(x) is not reciprocal then

$M(P) \ge M(x^3 -x - 1) \approx 1.3247$

and this is clearly best possible. If further all the coefficients of P are odd then

$M(P) \ge M(x^2 -x - 1) \approx 1.618 .$

If the field Q(α) is a Galois extension of Q then Lehmer's conjecture holds.

External links

• http://www.cecm.sfu.ca/~mjm/Lehmer/ is a nice reference about the problem.
• Weisstein, Eric W., "Lehmer's Mahler Measure Problem", MathWorld.