# 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.[1] The conjecture asserts that there is an absolute constant ${\displaystyle \mu >1}$ such that every polynomial with integer coefficients ${\displaystyle P(x)\in \mathbb {Z} [x]}$ satisfies one of the following properties:

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

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

${\displaystyle P(x)=a_{0}(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{D}),}$

and then set

${\displaystyle {\mathcal {M}}(P(x))=|a_{0}|\prod _{i=1}^{D}\max(1,|\alpha _{i}|).}$

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

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

for which the Mahler measure is the Salem number[2]

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

It is widely believed that this example represents the true minimal value: that is, ${\displaystyle \mu =1.176280818\dots }$ in Lehmer's conjecture.[3][4]

## Motivation

Consider Mahler measure for one variable and Jensen's formula shows that if ${\displaystyle P(x)=a_{0}(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{D})}$ then

${\displaystyle {\mathcal {M}}(P(x))=|a_{0}|\prod _{i=1}^{D}\max(1,|\alpha _{i}|).}$

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

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

Lehmer noticed[1][5] that ${\displaystyle m(P)=0}$ is an important value in the study of the integer sequences ${\displaystyle \Delta _{n}={\text{Res}}(P(x),x^{n}-1)=\prod _{i=1}^{D}(\alpha _{i}^{n}-1)}$ for monic ${\displaystyle P}$ . If ${\displaystyle P}$ does not vanish on the circle then ${\displaystyle \lim |\Delta _{n}|^{1/n}={\mathcal {M}}(P)}$ and this statement might be true even if ${\displaystyle P}$ does vanish on the circle. By this he was led to ask

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

or

given ${\displaystyle c>0}$, are there ${\displaystyle P}$ with integer coefficients for which ${\displaystyle 0?

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

Smyth [6] proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying ${\displaystyle x^{D}P(x^{-1})\neq P(x)}$.

Blanksby and Montgomery[7] and Stewart[8] independently proved that there is an absolute constant ${\displaystyle C>1}$ such that either ${\displaystyle {\mathcal {M}}(P(x))=1}$ or[9]

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

Dobrowolski [10] improved this to

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

Dobrowolski obtained the value C ≥ 1/1200 and asymptotically C > 1-ε for all sufficiently large D. Voutier in 1996 obtained C ≥ 1/4 for D ≥ 2.[11]

## Elliptic analogues

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

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

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

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

due to Laurent.[12] For arbitrary elliptic curves, the best known result is

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

due to Masser.[13] For elliptic curves with non-integral j-invariant, this has been improved to

${\displaystyle {\hat {h}}_{E}(Q)\geq {\frac {C(E/K)}{D^{2}(\log D)^{2}}},}$

by Hindry and Silverman.[14]

## Restricted results

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

If P(x) is not reciprocal then

${\displaystyle M(P)\geq M(x^{3}-x-1)\approx 1.3247}$

and this is clearly best possible.[15] If further all the coefficients of P are odd then[16]

${\displaystyle M(P)\geq M(x^{2}-x-1)\approx 1.618.}$

For any algebraic number α, let ${\displaystyle M(\alpha )}$ be the Mahler measure of the minimal polynomial ${\displaystyle P_{\alpha }}$ of α. If the field Q(α) is a Galois extension of Q, then Lehmer's conjecture holds for ${\displaystyle P_{\alpha }}$.[16]

## References

1. ^ a b Lehmer, D.H. (1933). "Factorization of certain cyclotomic functions". Ann. Math. (2). 34: 461–479. doi:10.2307/1968172. ISSN 0003-486X. Zbl 0007.19904.
2. ^ Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Springer-Verlag. p. 16. ISBN 0-387-95444-9. Zbl 1020.12001.
3. ^ Smyth (2008) p.324
4. ^ Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. 104. Providence, RI: American Mathematical Society. p. 30. ISBN 0-8218-3387-1. Zbl 1033.11006.
5. ^ David Boyd (1981). "Speculations concerning the range of Mahler's measure" Canad. Math. Bull. Vol. 24(4)
6. ^ Smyth, C. J. (1971). "On the product of the conjugates outside the unit circle of an algebraic integer". Bulletin of the London Mathematical Society. 3: 169–175. doi:10.1112/blms/3.2.169. Zbl 1139.11002.
7. ^ Blanksby, P. E.; Montgomery, H. L. (1971). "Algebraic integers near the unit circle". Acta Arith. 18: 355–369. Zbl 0221.12003.
8. ^ Stewart, C. L. (1978). "Algebraic integers whose conjugates lie near the unit circle". Bull. Soc. Math. France. 106: 169–176.
9. ^ Smyth (2008) p.325
10. ^ Dobrowolski, E. (1979). "On a question of Lehmer and the number of irreducible factors of a polynomial". Acta Arith. 34: 391–401.
11. ^ P. Voutier, An effective lower bound for the height of algebraic numbers, Acta Arith. 74 (1996), 81–95.
12. ^ Smyth (2008) p.327
13. ^ Masser, D.W. (1989). "Counting points of small height on elliptic curves". Bull. Soc. Math. Fr. 117 (2): 247–265. Zbl 0723.14026.
14. ^ Hindry, Marc; Silverman, Joseph H. (1990). "On Lehmer's conjecture for elliptic curves". In Goldstein, Catherine. Sémin. Théor. Nombres, Paris/Fr. 1988-89. Prog. Math. 91. pp. 103–116. ISBN 0-8176-3493-2. Zbl 0741.14013.
15. ^ Smyth (2008) p.328
16. ^ a b Smyth (2008) p.329
• Smyth, Chris (2008). "The Mahler measure of algebraic numbers: a survey". In McKee, James; Smyth, Chris. Number Theory and Polynomials. London Mathematical Society Lecture Note Series. 352. Cambridge University Press. pp. 322–349. ISBN 978-0-521-71467-9.