Lehmer's conjecture

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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 such that every polynomial with integer coefficients satisfies one of the following properties:

  • The Mahler measure of is greater than or equal to .
  • is an integral multiple of a product of cyclotomic polynomials or the monomial , in which case . (Equivalently, every complex root of is a root of unity or zero.)

There are a number of definitions of the Mahler measure, one of which is to factor over as

and then set

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

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

It is widely believed that this example represents the true minimal value: that is, in Lehmer's conjecture.[3][4]

Motivation[edit]

Consider Mahler measure for one variable and Jensen's formula shows that if then

In this paragraph denote  , which is also called Mahler measure.

If has integer coefficients, this shows that is an algebraic number so is the logarithm of an algebraic integer. It also shows that and that if then is a product of cyclotomic polynomials i.e. monic polynomials whose all roots are roots of unity, or a monomial polynomial of i.e. a power for some .

Lehmer noticed[1][5] that is an important value in the study of the integer sequences for monic . If does not vanish on the circle then and this statement might be true even if does vanish on the circle. By this he was led to ask

whether there is a constant such that provided is not cyclotomic?,

or

given , are there with integer coefficients for which ?

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[edit]

Let be an irreducible monic polynomial of degree .

Smyth [6] proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying .

Blanksby and Montgomery[7] and Stewart[8] independently proved that there is an absolute constant such that either or[9]

Dobrowolski [10] improved this to

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[edit]

Let be an elliptic curve defined over a number field , and let be the canonical height function. The canonical height is the analogue for elliptic curves of the function . It has the property that if and only if is a torsion point in . The elliptic Lehmer conjecture asserts that there is a constant such that

for all non-torsion points ,

where . If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds:

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

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

by Hindry and Silverman.[14]

Restricted results[edit]

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

If P(x) is not reciprocal then

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


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

References[edit]

  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. 

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