Lehmer conjecture

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].

References

[a1]	P.E. Blanksby, H.L. Montgomery, "Algebraic integers near the unit circle" Acta Arith. , 18 (1971) pp. 355–369
[a2]	K. Breusch, "On the distribution of the roots of a polynomial with integral coefficients" Proc. Amer. Math. Soc. , 3 (1951) pp. 939–941
[a3]	D.G. Cantor, E.G. Straus, "On a conjecture of D.H. Lehmer" Acta Arith. , 42 (1982) pp. 97–100; 325
[a4]	E. Dobrowolski, "On a question of Lehmer and the number of irreducible factors of a polynomial" Acta Arith. , 34 (1979) pp. 391–401
[a5]	A. Dubickas, "On algebraic numbers of small measure" Liet. Mat. Rink. , 35 (1995) pp. 421–431
[a6]	A. Dubickas, "Algebraic conjugates outside the unit circle" , New Trends in Probability and Statistics , 4 (1997) pp. 11–21
[a7]	D.H. Lehmer, "Factorization of certain cyclotomic functions" Ann. Math. , 34 : 2 (1933) pp. 461–479
[a8]	D.A. Lind, K. Schmidt, T. Ward, "Mahler measure and entropy for commuting automorphisms of compact groups" Invent. Math. , 101 (1990) pp. 503–629
[a9]	R. Louboutin, "Sur la mesure de Mahler d'un nombre algébrique" C.R. Acad. Sci. Paris , 296 (1983) pp. 707–708
[a10]	K. Mahler, "An application of Jensen's formula to polynomials" Mathematika , 7 (1960) pp. 98–100
[a11]	K. Mahler, "On some inequalities for polynomials in several variables" J. London Math. Soc. , 37 (1962) pp. 341–344
[a12]	U. Rausch, "On a theorem of Dobrowolski about the product of conjugate numbers" Colloq. Math. , 50 (1985) pp. 137–142
[a13]	A. Schinzel, "The Mahler measure of polynomials" , Number Theory and its Applications (Ankara, 1996) , M. Dekker (1999) pp. 171–183
[a14]	C.J. Smyth, "On the product of the conjugates outside the unit circle of an algebraic integer" Bull. London Math. Soc. , 3 (1971) pp. 169–175
[a15]	C.L. Stewart, "Algebraic integers whose conjugates lie near the unit circle" Bull. Soc. Math. France , 196 (1978) pp. 169–176
[a16]	A. Schinzel, H. Zassenhaus, "A refinement of two theorems of Kronecker" Michigan J. Math. , 12 (1965) pp. 81–85