Lehmer’s Conjecture: Mahler Measure Bounds

• Lehmer’s Conjecture is a hypothesis asserting that every noncyclotomic monic integer polynomial has a Mahler measure uniformly bounded away from 1, with Lehmer’s number (~1.17628) serving as the benchmark.
• The conjecture connects number theory and geometry, impacting short geodesic lengths in arithmetic hyperbolic orbifolds and height claims via Dobrowolski’s theorem.
• Extensions to p-adic, group-theoretic, and elliptic curve contexts demonstrate the conjecture’s broad implications, while its claimed proofs remain unverified and the problem open.

Lehmer’s conjecture, in the standard number-theoretic sense, is the assertion that the Mahler measure of a noncyclotomic monic integer polynomial is bounded away from $1$ by an absolute constant. Equivalently, for every nonzero algebraic integer $\beta$ that is not a root of unity, there should exist a universal constant $C>1$ such that $M(\beta)\ge C$. The problem originates in Lehmer’s 1933 question asking whether, for every $\varepsilon>0$, one can find an algebraic integer $\beta$ with $1<M(\beta)<1+\varepsilon$; the conjecture is the modern “negative answer” to that question. In the literature considered here, this is distinct from the separate conjectures on Ramanujan’s tau function and on Euler’s totient function that also bear Lehmer’s name (Amoroso, 2018).

1. Definition and equivalent formulations

If

$P(x)=\prod_{i=1}^n (x-\theta_i)$

is an irreducible monic polynomial with integer coefficients, its Mahler measure is

$M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$

For a nonzero algebraic integer $\beta$ with minimal polynomial $\beta$0, one writes $\beta$1; in the algebraic-integer formulation used by Amoroso, this is the product of the absolute values of the conjugates of $\beta$2 outside the unit circle (Belolipetsky, 2011, Amoroso, 2018).

Kronecker’s theorem gives the basic dichotomy: $\beta$3 Accordingly, Lehmer’s conjecture asks for a uniform gap between $\beta$4 and the Mahler measures of all nontrivial algebraic integers. In polynomial language, the conjecture is commonly stated as: there exists $\beta$5 such that $\beta$6 for every non-cyclotomic monic integer polynomial $\beta$7 (Belolipetsky, 2011).

The height-theoretic formulation is equivalent. For an algebraic number $\beta$8,

$\beta$9

so a Lehmer-type lower bound is a statement that $C>1$0 cannot decay faster than order $C>1$1 for non-torsion $C>1$2. This is the form used in later $C>1$3-adic and equivariant generalisations (Silverman, 2010, Dixit et al., 27 Jul 2025).

2. Classical lower bounds and the reciprocal obstruction

The conjecture remains unsolved. Amoroso explicitly recalls that the best unconditional result is Dobrowolski’s theorem: for every $C>1$4 there exists $C>1$5 such that

$C>1$6

for algebraic integers $C>1$7 of degree $C>1$8 that are not roots of unity (Amoroso, 2018). In geometric applications, the same input appears in the more familiar form

$C>1$9

which Belolipetsky uses to quantify how slowly arithmetic hyperbolic systoles could tend to zero (Belolipetsky, 2011).

Several reductions clarify where the difficulty lies. If $M(\beta)\ge C$0 is not an algebraic integer, then $M(\beta)\ge C$1. If $M(\beta)\ge C$2 is an algebraic integer with nonreciprocal minimal polynomial, then Smyth’s theorem gives

$M(\beta)\ge C$3

where $M(\beta)\ge C$4 is the smallest Pisot number. Thus the delicate regime is the reciprocal case, where one seeks to exclude Mahler measures accumulating just above $M(\beta)\ge C$5 (Verger-Gaugry, 2019).

The benchmark remains Lehmer’s degree-$M(\beta)\ge C$6 example

$M(\beta)\ge C$7

whose Mahler measure is

$M(\beta)\ge C$8

No smaller Mahler measure has been found for a noncyclotomic monic integer polynomial, and $M(\beta)\ge C$9 is accordingly the expected optimal lower bound in many restricted families (Taylor, 2011).

There are also explicit positive results for special congruence-divisibility classes. Silverman proved that if $\varepsilon>0$0 is monic, none of its roots is a root of unity, and $\varepsilon>0$1 divides $\varepsilon>0$2 in $\varepsilon>0$3 for some

$\varepsilon>0$4

then

$\varepsilon>0$5

In particular, Lehmer’s conjecture holds for that class of polynomials (Silverman, 2010).

3. Families for which the Lehmer bound is known

A substantial body of work proves the conjectural lower bound for special families of reciprocal polynomials arising from matrices. Let $\varepsilon>0$6 be a Hermitian matrix over an imaginary quadratic integer ring, with characteristic polynomial $\varepsilon>0$7, and consider the associated reciprocal polynomial $\varepsilon>0$8. In this setting the relevant dichotomy is: $\varepsilon>0$9 Thus Lehmer’s number is attained as the lower threshold for these matrix-associated families (Taylor, 2011, Greaves et al., 2012).

Taylor proved this for Hermitian matrices over the rings of integers of imaginary quadratic fields with $\beta$0, squarefree, $\beta$1, and McKee–Smyth had earlier handled integer symmetric matrices. The Gaussian and Eisenstein cases were then completed by McKee and Yatsyna, who proved the same lower bound for Hermitian matrices over $\beta$2 and $\beta$3. In both papers the proof proceeds by translating Hermitian matrices into charged weighted graphs, classifying cyclotomic graphs, reducing to minimal non-cyclotomic configurations via interlacing, and then performing a finite obstruction search (Taylor, 2011, Greaves et al., 2012).

These results matter because reciprocal polynomials are the hard part of Lehmer’s problem. As one of the matrix papers stresses, Smyth’s theorem already shows that the smallest Mahler measure of a non-reciprocal monic integer polynomial is $\beta$4, so any full resolution of Lehmer’s conjecture must control reciprocal structure in a way that the matrix classification achieves only for special classes (Greaves et al., 2012).

4. Geometric reformulations and consequences

A striking reformulation links Lehmer’s conjecture to short geodesics in arithmetic hyperbolic orbifolds. If $\beta$5 is a hyperbolic element of $\beta$6 or $\beta$7 with

$\beta$8

and $\beta$9 is the minimal polynomial of $1<M(\beta)<1+\varepsilon$0, then the displacement length satisfies

$1<M(\beta)<1+\varepsilon$1

and

$1<M(\beta)<1+\varepsilon$2

For arithmetic lattices, $1<M(\beta)<1+\varepsilon$3 is an algebraic integer and $1<M(\beta)<1+\varepsilon$4 is non-cyclotomic, so lower bounds for Mahler measure become lower bounds for lengths of closed geodesics (Belolipetsky, 2011).

Belolipetsky formulates the corresponding “Short Geodesic Conjecture”: there should be a universal positive lower bound for geodesic lengths in arithmetic hyperbolic $1<M(\beta)<1+\varepsilon$5- and $1<M(\beta)<1+\varepsilon$6-orbifolds. He records the precise logical relation:

1. Lehmer’s conjecture implies the Short Geodesic Conjecture.
2. The Short Geodesic Conjecture implies Lehmer’s conjecture for Salem numbers. So the converse is only partial: the geometric statement recovers a Salem-number case, not full Lehmer (Belolipetsky, 2011).

The same paper gives quantitative consequences of hypothetical failure. If a sequence $1<M(\beta)<1+\varepsilon$7 of arithmetic orbifolds had

$1<M(\beta)<1+\varepsilon$8

then necessarily

$1<M(\beta)<1+\varepsilon$9

and combining Dobrowolski’s bound with field-degree and volume estimates yields a lower bound of the form

$P(x)=\prod_{i=1}^n (x-\theta_i)$0

This tends to $P(x)=\prod_{i=1}^n (x-\theta_i)$1 extraordinarily slowly, showing that any arithmetic short-geodesic counterexample would have to occur in an extreme large-volume, high-degree regime (Belolipetsky, 2011).

5. Height-theoretic, $P(x)=\prod_{i=1}^n (x-\theta_i)$2-adic, and group-theoretic extensions

Modern work often treats Lehmer’s conjecture as the prototype for a broader family of height lower bounds. One such direction is a $P(x)=\prod_{i=1}^n (x-\theta_i)$3-adic criterion. Fix a finite extension $P(x)=\prod_{i=1}^n (x-\theta_i)$4 with residue field $P(x)=\prod_{i=1}^n (x-\theta_i)$5, and let $P(x)=\prod_{i=1}^n (x-\theta_i)$6 denote the set of conjugates of $P(x)=\prod_{i=1}^n (x-\theta_i)$7 lying in $P(x)=\prod_{i=1}^n (x-\theta_i)$8. Dubickas and Mossinghoff prove

$P(x)=\prod_{i=1}^n (x-\theta_i)$9

where $M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$0. As a consequence, if

$M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$1

then Lehmer’s conjecture holds for $M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$2 (Dixit et al., 27 Jul 2025).

Another direction replaces the ordinary Weil height by a finite Möbius-orbit height. For a finite subgroup $M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$3, van Ittersum defines

$M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$4

If the set of $M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$5-orbits all of whose nonzero elements lie on the unit circle is finite, then there exists $M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$6 such that

$M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$7

for all algebraic $M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$8. This gives a genuine group-invariant Lehmer theorem, recovering earlier results of Zagier and Dresden as special cases (Ittersum, 2016).

A further generalisation concerns finite-rank subgroups of semiabelian varieties. Checcoli and Dill study a consequence of Rémond’s generalisation of Lehmer’s conjecture: for an almost split semiabelian variety $M(P)=\prod_{i=1}^n \max(1,|\theta_i|).$9, a finite-rank subgroup $\beta$0, and a finite extension

$\beta$1

they prove that

$\beta$2

is free abelian. Their point is not a direct height lower bound, but a structural property predicted by the conjectural lower-bound picture (Checcoli et al., 23 Jun 2025).

Experimental work on elliptic curves makes the analogy explicit. For a fixed elliptic curve $\beta$3, the elliptic Lehmer conjecture asks whether

$\beta$4

over non-torsion points $\beta$5. Recent computations over quadratic fields build finite-search methods for points of very small canonical height and use the resulting data to test Lehmer- and Lang-type expectations, without proving a general lower bound (Cats et al., 10 Oct 2025).

6. Claimed proofs, corrections, and present status

The conjecture has generated several announced proofs, but the accepted status remains open. A notable 2017 preprint, “A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus,” claimed a dynamical proof based on Parry upper functions $\beta$6, dynamical zeta functions of Rényi–Parry systems, a “lenticular” set of poles of $\beta$7, and asymptotic expansions in the dynamical degree. It asserted, among other things, a universal lower bound

$\beta$8

for nonzero algebraic integers that are not roots of unity (Verger-Gaugry, 2017).

Amoroso’s corrective note explained why that announced proof is not valid. The issue occurs in the treatment of

$\beta$9

which is only known to be meromorphic on the open unit disk. The preprint considers simple zeros $\beta$00 of $\beta$01, differentiates the formal identity

$\beta$02

and then substitutes $\beta$03 to conclude that $\beta$04 is not a pole of $\beta$05. Amoroso points out that this substitution is illegitimate, because whether $\beta$06 has a pole at $\beta$07 is exactly the point at issue; if $\beta$08 has a pole there, the differentiated identity cannot simply be evaluated as though all terms were finite. He adds that the same type of argument is used elsewhere in the preprint (Amoroso, 2018).

A later 2019 preprint again announced a proof of Lehmer’s conjecture through Parry upper functions, lenticuli of poles, and dynamical-degree asymptotics, together with related claims on Schinzel–Zassenhaus, Salem numbers, and equidistribution (Verger-Gaugry, 2019). The practical consequence of Amoroso’s note, however, is unchanged: the 2017 claimed proof should not be regarded as established mathematics, and Lehmer’s conjecture remains open (Amoroso, 2018).

The modern state of the problem is therefore sharply stratified. There are strong general lower bounds, exact results for several natural families, geometric and dynamical reformulations, and increasingly refined local criteria, but no accepted proof of a universal constant $\beta$09 valid for all nonzero algebraic integers that are not roots of unity.