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

The 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).

What gets accepted

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