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

• Nonreciprocal case (Smyth). Smyth answered Lehmer’s question for nonreciprocal polynomials: if $f\in\mathbb{Z}[x]$ is nonreciprocal and $f(0)\neq 0$, then

[BDM2007-smyth-thm1] [Smy1971]

References (from the repository page)

• [BDM2007] Borwein, Peter; Dobrowolski, Edward; Mossinghoff, Michael J. Lehmer’s problem for polynomials with odd coefficients. Annals of Mathematics 166(2) (2007), 347–366. DOI: 10.4007/annals.2007.166.347. Google Scholar. Author PDF

• [BDM2007-def-M] loc: Author PDF p.1, Equation (1.1). quote: “Mahler’s measure of a polynomial $f$, denoted $M(f)$, is defined as the product of the absolute values of those roots of $f$ that lie outside the unit disk, multiplied by the absolute value of the leading coefficient. Writing $f(x)=a\prod_{k=1}^d (x-\alpha_k)$, we have $M(f)=|a|\prod_{k=1}^d \max\{1,|\alpha_k|\}$.”

• [BDM2007-kronecker] loc: Author PDF p.2, Introduction (paragraph after Equation (1.1)). quote: “For $f \in \mathbb{Z}[x]$, clearly $M(f) \ge 1$, and by a classical theorem of Kronecker, $M(f) = 1$ precisely when $f(x)$ is a product of cyclotomic polynomials and the monomial $x$.”

• [BDM2007-lehmer-question] loc: Author PDF p.2, Introduction (Lehmer’s question). quote: “In 1933, D. H. Lehmer [12] asked if for every $\epsilon > 0$ there exists a polynomial $f \in \mathbb{Z}[x]$ satisfying $1 < M(f) < 1 + \epsilon$.”

• [BDM2007-lehmer-poly] loc: Author PDF p.2, Introduction (Lehmer’s example). quote: “Lehmer noted that the polynomial $\ell(x)=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$ has $M(\ell)=1.176280\ldots$, and this value remains the smallest known measure larger than $1$ of a polynomial with integer coefficients.”

• [BDM2007-smallest-known] loc: Author PDF p.2, Introduction (after Lehmer’s example). quote: “and this value remains the smallest known measure larger than $1$ of a polynomial with integer coefficients.”

• [BDM2007-smyth-thm1] loc: Author PDF p.2, Introduction (Smyth’s result). quote: “Smyth [22] showed that if $f \in \mathbb{Z}[x]$ is nonreciprocal and $f(0) \ne 0$, then $M(f) \ge M(x^3-x-1) = 1.324717 \ldots$.”

• [Leh1933] Lehmer, D. H. Factorization of Certain Cyclotomic Functions. Annals of Mathematics 34(3) (1933), 461–479. DOI: 10.2307/1968172. Google Scholar.

• [Smy1971] Smyth, C. J. On the product of the conjugates outside the unit circle of an algebraic integer. Bulletin of the London Mathematical Society 3(2) (1971), 169–175. DOI: 10.1112/blms/3.2.169. Google Scholar.

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1; best known upper bound 1.176280.... Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds