On Lehmer's Problem
Abstract
Let m≥ 2 be an integer, and let d≥ 2 be an even integer. We study monic, reciprocal, non-cyclotomic polynomials P(x)=xd+ad−1xd−1+⋯+a1x+1∈ℤ[x]. For a fixed residue class r∈{0,1,…,m−1}, we prove that there exists an effectively computable constant cr,m>1 such that the Mahler measure M(P)≥ cr,m for every irreducible polynomial Q whose inner coefficients satisfy aj≡ rpmod m (1≤ j≤ d−1). Moreover, we explicitly compute the constant cr,m in terms of r and m. For the residue classes r=2, r=3, and r=m−1, we obtain improved lower bounds for suitable ranges of the modulus~m. We also consider monic, reciprocal, irreducible, non-cyclotomic polynomials whose inner coefficients belong to two distinct residue classes modulo m. More precisely, if the inner coefficients with odd indices are congruent to apmod m and those with even indices are congruent to bpmod m, where anot≡ bpmod m, then we prove the existence of a constant cm,a,b>1 such that the Mahler measure M(P)≥ cm,a,b. These results solve Lehmer's problem for several families of polynomials. We also construct infinitely many examples in every even degree d≥ 2 of monic, reciprocal, irreducible, non-cyclotomic polynomials over ℚ, together with explicit information about their Mahler measures. Our proofs rely on Y. Bilu's equidistribution theorem and its effective version proved by C. D'Andrea \emph{et al.}, as well as on a method developed by P. Borwein \emph{et al.} While it is natural to consider Bilu's equidistribution results in conjunction with the Borwein {\it et al}. method to Lehmer's problem, we are not aware of any previous work in the literature that employs this particular approach.
Related
BibTeX
@misc{jha2026lehmers,
title = {{On Lehmer's Problem}},
author = {Piyush Kumar Jha and R. Thangadurai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33631},
eprint = {2610.33631},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33631},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Lehmer's Problem
AU - Piyush Kumar Jha
AU - R. Thangadurai
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33631
UR - https://arxiv.org/abs/2610.33631
AB - Let m≥ 2 be an integer, and let d≥ 2 be an even integer. We study monic, reciprocal, non-cyclotomic polynomials P(x)=x^d+a_d−1x^d−1+⋯+a_1x+1∈ℤ[x]. For a fixed residue class r∈{0,1,…,m−1}, we prove that there exists an effectively computable constant c_r,m>1 such that the Mahler measure M(P)≥ c_r,m for every irreducible polynomial Q whose inner coefficients satisfy a_j≡ rpmod m (1≤ j≤ d−1). Moreover, we explicitly compute the constant c_r,m in terms of r and m. For the residue classes r=2, r=3, and r=m−1, we obtain improved lower bounds for suitable ranges of the modulus~m. We also consider monic, reciprocal, irreducible, non-cyclotomic polynomials whose inner coefficients belong to two distinct residue classes modulo m. More precisely, if the inner coefficients with odd indices are congruent to apmod m and those with even indices are congruent to bpmod m, where anot≡ bpmod m, then we prove the existence of a constant c_m,a,b>1 such that the Mahler measure M(P)≥ c_m,a,b. These results solve Lehmer's problem for several families of polynomials. We also construct infinitely many examples in every even degree d≥ 2 of monic, reciprocal, irreducible, non-cyclotomic polynomials over ℚ, together with explicit information about their Mahler measures. Our proofs rely on Y. Bilu's equidistribution theorem and its effective version proved by C. D'Andrea \emph{et al.}, as well as on a method developed by P. Borwein \emph{et al.} While it is natural to consider Bilu's equidistribution results in conjunction with the Borwein {\it et al}. method to Lehmer's problem, we are not aware of any previous work in the literature that employs this particular approach.
ER -