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on Lehmer’s Mahler measure constant ($C_{40a}$)

The statement

Let

f(x)=\sum_{i=0}^n a_i x^i \;=\; a_n\prod_{i=1}^n (x-\alpha_i)

be a polynomial with complex coefficients. The Mahler measure of $f$ is

M(f)\;:=\;|a_n|\prod_{i=1}^n \max\{1,|\alpha_i|\}.

[BDM2007-def-M]

For an integer polynomial $f(x)\in\mathbb{Z}[x]$ , Kronecker’s theorem characterizes the case $M(f)=1$ :

M(f)=1 \quad\Longleftrightarrow\quad f(x)\text{ is a product of cyclotomic polynomials and }x.

[BDM2007-kronecker]

Motivated by Lehmer’s question, define Lehmer’s Mahler measure constant $C_{40a}$ to be the infimum of Mahler measures strictly larger than $1$ among integer polynomials, and denote it by $L$ :

L \;:=\; \inf\bigl\{ M(f)\;:\; f\in\mathbb{Z}[x],\ 1<M(f)\bigr\}.

[BDM2007-lehmer-question]

Lehmer’s original question (1933) asks whether, for every $\epsilon>0$ , there exists an integer polynomial $f$ with

1<M(f)<1+\epsilon,

which is equivalent to asking whether $L=1$ . [BDM2007-lehmer-question]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.