Lehmer's conjecture

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

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, in which case $\mathcal{M}(P(x))=1$. (Equivalently, every complex root of $P(x)$ is a root of unity.)

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

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

The smallest known Mahler measure (greater than 1) is for the polynomial

$P(x)= x^{10}-x^9+x^7-x^6+x^5-x^4+x^3-x+1$, which satisfies $\mathcal{M}(P(x))=1.176280818\dots\,.$

It is widely believed that this example represents the true minimal value.

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

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

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

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

External links

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