Lehmer’s conjecture is true for polynomials with odd coefficients

You need to know: Polynomials, degree of a polynomial, roots of a polynomial, divisor of a polynomial, irreducible polynomial, complex numbers, notation ${\mathbb Z}[x]$ for polynomials in variable x with integer coefficients, notation $a \equiv b\,(\text{mod}\, m)$ if $a-b$ is divisible by m.

Background: An irreducible polynomial $P \in {\mathbb Z}[x]$ is cyclotomic if $P$ is a divisor of $x^n-1$ for some integer $n \geq 1$ . Every polynomial $P \in {\mathbb Z}[x]$ of degree n can be written as $P(x)=a\prod\limits_{i=1}^n(x-\alpha_i)$ , where $a\in {\mathbb Z}$ and $\alpha_i$ are (possibly complex) roots of P. Mahler ’s measure of P is $M(P):=|a|\prod\limits_{i=1}^n\max\{1,|\alpha_i|\}$ .

For integer $m\geq 2$ , let $D_m:=\left\{\sum\limits_{i=0}^n a_ix^i \in {\mathbb Z}[x]\,|\,a_i \equiv 1 (\text{mod}\, m), \, 0\leq i \leq n\right\}$ .

The Theorem: On 2nd October 2003, Peter Borwein, Edward Dobrowolski, and Michael Mossinghoff submitted to the Annals of Mathematics a paper in which they proved that inequality $\log M(P) \geq c_m\left(1-\frac{1}{n+1}\right)$ holds for every $P \in D_m$ of degree n and no cyclotomic divisors, where $c_2=(\log 5)/4$ and $c_m=\log(\sqrt{m^2+1}/2)$ for $m>2$ .

Short context: In 1933, Lehmer asked if for every $\epsilon>0$ there exists a polynomial $P \in {\mathbb Z}[x]$ satisfying $1<M(P)<1+\epsilon$ . It is conjectured that the answer to this question is negative, and this is known as Lehmer’s conjecture. The Theorem implies that this conjecture holds for polynomials $P \in D_m$ . In particular, case $m=2$ of the Theorem implies that Lehmer’s conjecture holds for polynomials with odd coefficients.