OP-12051OpenNumber theory

Asymptotic Dobrowolski constant for Lehmer’s problem ( $C_{40b}$ )

Asymptotic Dobrowolski constant for Lehmer’s problem: determine $C_{40b}$ or improve its bounds; the repository records $9/4$ (lower) and $\infty$ (upper).

MSC 11R06 · PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

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Problem statement

Let $\alpha$ be a nonzero algebraic number of degree $d$ , with minimal polynomial over $\mathbb{Z}$

f(X)=a_d\prod_{i=1}^d (X-\alpha_i),

where $a_d>0$ and $\alpha_1,\dots,\alpha_d$ are the conjugates of $\alpha$ . Define the Mahler measure of $\alpha$ by

M(\alpha)\;:=\;a_d\prod_{i=1}^d \max\{1,\lvert \alpha_i\rvert\}.

[Vou1996-def-M]

Define the absolute logarithmic height $h(\alpha)$ by

h(\alpha)\;:=\;\frac{\log M(\alpha)}{d}.

[Vou1996-def-h]

Write

B(d)\;:=\;\left(\frac{\log\log d}{\log d}\right)^3 \qquad (d\ge 3),

and consider algebraic numbers $\alpha$ that are not roots of unity. Dobrowolski proved an asymptotic lower bound of the form

M(\alpha)\;>\;1+(1-\epsilon)\,B(d)\quad \text{for }d\ge d(\epsilon),

for each $\epsilon>0$ . [Vou1996-dob-asymp]

Motivated by this asymptotic form, define the asymptotic Dobrowolski constant $C_{40b}$ to be the largest constant $c$ such that, for every $\epsilon>0$ , there exists $d(\epsilon)$ with

M(\alpha)\;\ge\;1+(c-\epsilon)\,B(d)\quad \text{for all non-root-of-unity }\alpha\text{ of degree }d\ge d(\epsilon).

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Background

$M(\alpha)$ vs. $\log M(\alpha)$ . Many statements in the literature are formulated for $\log M(\alpha)$ (equivalently $d\,h(\alpha)$ ). Voutier’s inequality

1+\log(M(\alpha))<M(\alpha)

shows that any bound of the form $\log M(\alpha) > c\,B(d)$ immediately implies the corresponding bound $M(\alpha) > 1+c\,B(d)$ . [Vou1996-log-vs-M]

Earlier degree-dependent lower bounds of different shape. Before Dobrowolski’s $(\log\log d/\log d)^3$ term, Blanksby–Montgomery proved

M(\alpha)\;>\;1+\frac{1}{52d\log(6d)},

and Stewart proved

M(\alpha)\;>\;1+\frac{1}{10^4d\log(d)}.

[Vou1996-bm] [Vou1996-stew]

References (from the repository page)

[Vou1996] Voutier, Paul M. An effective lower bound for the height of algebraic numbers. Acta Arithmetica 74(1) (1996), 81–95. DOI: 10.4064/aa-74-1-81-95. Google Scholar. arXiv PDF.
[Vou1996-def-M] loc: arXiv v1 PDF p.1, Introduction (definition of $M(\alpha)$ ). quote: “We shall define the Mahler measure of $\alpha$ , $M(\alpha)$ , by $M(\alpha)=a_d\prod_{i=1}^d\max(1,\lvert\alpha_i\rvert)$ .”
[Vou1996-def-h] loc: arXiv v1 PDF p.1, Introduction (definition of $h(\alpha)$ ). quote: “ $h(\alpha)=\frac{\log M(\alpha)}{d}$ .”
[Vou1996-bm] loc: arXiv v1 PDF p.1, Introduction (Blanksby–Montgomery). quote: “They proved that $M(\alpha)>1+\frac{1}{52d\log(6d)}$ .”
[Vou1996-stew] loc: arXiv v1 PDF p.1, Introduction (Stewart). quote: “In 1978, C.L. Stewart [18] introduced a method from transcendental number theory to prove that $M(\alpha) > 1+1/(10^4d \log(d))$ .”
[Vou1996-dob-asymp] loc: arXiv v1 PDF p.2, Introduction (Dobrowolski’s asymptotic bound). quote: “Dobrowolski… showed that $M(\alpha)>1+(1-\epsilon)\left(\frac{\log\log d}{\log d}\right)^3$ for $d\ge d(\epsilon)$ .”
[Vou1996-cs-lou] loc: arXiv v1 PDF p.2, Introduction (Cantor–Straus; Louboutin). quote: “Cantor and Straus… replace the coefficient $(1-\epsilon)$ by $(2-\epsilon)$ . Louboutin… to $(\frac94-\epsilon)$ .”
[Vou1996-log-vs-M] loc: arXiv v1 PDF p.3, paragraph after Theorem. quote: “Notice that $1+\log(M(\alpha))<M(\alpha)\le\lvert\alpha\rvert^d$ .”
[Dob1979] Dobrowolski, E. On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arithmetica 34 (1979), 391–401. Google Scholar.
[CS1982] Cantor, D.; Straus, E. G. On a conjecture of D. H. Lehmer. Acta Arithmetica 42(1) (1982), 97–100. Google Scholar.
[Lou1983] Louboutin, R. Sur la mesure de Mahler d'un nombre algebrique. C. R. Acad. Sci. Paris Ser. I 296 (1983), 707–708. Google Scholar.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound $9/4$ ; best known upper bound $\infty$ . Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
$\infty$

Known lower bounds

Bound	Reference	Comments
$1$	[Dob1979]	Dobrowolski proved $M(\alpha) > 1+(1-\epsilon)B(d)$ for $d\ge d(\epsilon)$ (as reported by Voutier), hence $C_{40b}\ge 1$ . [Vou1996-dob-asymp]
$2$	[CS1982]	Cantor–Straus replace the coefficient $(1-\epsilon)$ by $(2-\epsilon)$ (as reported by Voutier), hence $C_{40b}\ge 2$ . [Vou1996-cs-lou]
$9/4$	[Lou1983]	Louboutin improves the coefficient to $(\tfrac94-\epsilon)$ (as reported by Voutier), hence $C_{40b}\ge 9/4$ . [Vou1996-cs-lou]

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Asymptotic Dobrowolski constant for Lehmer’s problem ($C_{40b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/40b.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Asymptotic Dobrowolski constant for Lehmer’s problem ($C_{40b}$)
Added: 5 Nov 2026
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Number theory