# Lehmer conjecture

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A conjecture about the minimal Mahler measure of a non-zero algebraic integer which is not a root of unity. The Mahler measure $M(\alpha)$ of an algebraic number $\alpha$ is defined by $$M(\alpha) = a_0 \prod_{i=1}^N \max(1,|\alpha_i|) \ ,$$ where $a_0$ denotes the leading coefficient and $N$ the degree of the minimal polynomial $f$ (with integral coefficients) of $\alpha$ (cf. also Algebraic number) and $\alpha=\alpha_1,\alpha_2,\ldots,\alpha_N$ are its conjugates. Since $M(\alpha)$ depends only on $f$, it is also denoted by $M(f)$ and called the Mahler measure of $f$. Jensen's formula (cf. also Jensen formula) implies the equality $$M(f) = \exp\left({ \int_0^1 \log\left\vert{ f\left({ e^{2\pi i t} }\right) }\right\vert d t }\right)$$ and this observation permits one to generalize Mahler's measure to polynomials in several variables (see [a11], [a13]).

A theorem of L. Kronecker implies that if $\alpha$ is an algebraic integer with $M(\alpha)\le1$, then $\alpha$ is either zero or a root of unity. D.H. Lehmer [a7] asked whether $M(\alpha)$ could attain values arbitrarily close to $1$. This subsequently led to the following formulation of Lehmer's conjecture: There exists a positive constant $\eta$ such that if $\alpha\ne0$ is an algebraic integer, not a root of unity, then $$M(\alpha) \ge 1 + \eta \ .$$

Lehmer's conjecture is equivalent to the existence of ergodic automorphisms of the infinite-dimensional torus having finite entropy [a8] and its truth would imply the following conjecture stated by A. Schinzel and H. Zassenhaus [a16]: There exists a positive constant $C$ with the property that if $\alpha$ is a non-zero algebraic integer of degree $N$, not a root of unity, then $\boxed{\alpha}$, the maximal absolute value of a conjugate of $\alpha$ is at least $$1+\frac{C}{N}$$

It is known ([a2], [a14]) that Lehmer's conjecture holds for non-reciprocal integers $\alpha$, i.e. algebraic integers whose minimal polynomials do not have $1/\alpha$ as a root. In this case the minimal value for $M(\alpha)$ equals $1.32471\ldots$ and is attained by roots of the polynomial $X^3-X-1$.

In 1971, P.E. Blanksby and H.L. Montgomery [a1] established, for all algebraic integers $\alpha\ne0$ of degree $N$ that are not roots of unity, the inequality $$M(\alpha) \ge 1 + \frac{ 1 }{ 52N\log(6N) }$$ and subsequently E. Dobrowolski [a4] obtained $$M(\alpha) > 1 + c\left({ \frac{ \log\log N }{ \log N } }\right)^3$$ with $c=1/1200$, whereas for $N \ge N(\epsilon)$ he got $c=1-\epsilon$. Subsequently, several authors increased the value of $c$ to $c=2-\epsilon$ ([a3], [a12]) and $c=9/4-\epsilon$ ([a9]). Since for non-reciprocal integers $\alpha$ one has $M(\alpha) \le \boxed{\alpha}^{N/2}$, the last result leads to the inequality $$\boxed{\alpha} > 1 + \frac{ 9 }{ 2N } \left({ \frac{ \log\log N }{ \log N } }\right)^3$$ but this has been superseded by A. Dubickas [a5], who proved for sufficiently large $N$ the inequality $$\boxed{\alpha} > 1 + \left({ \frac{64}{\pi^2}-\epsilon }\right) \left({ \frac{ \log\log N }{ \log N } }\right)^3$$ which is the strongest known result toward the Schinzel–Zassenhaus conjecture as of 2000.

The smallest known value of $M(\alpha) > 1$ is$1.17628\ldots$, achieved by the root of $X^{10} + X^9 - X^7 -X^6 - X^5 - X^4 - X^3 + X + 1$ and found in [a7].

How to Cite This Entry:
Lehmer conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lehmer_conjecture&oldid=43030
This article was adapted from an original article by WÅ‚adysÅ‚aw Narkiewicz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article