# The Lehmer constants of an annulus

• Volume: 13, Issue: 2, page 413-420
• ISSN: 1246-7405

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## Abstract

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Let $M\left(\alpha \right)$ be the Mahler measure of an algebraic number $\alpha$, and $V$ be an open subset of $ℂ$. Then its Lehmer constant$L\left(V\right)$ is inf $M{\left(\alpha \right)}^{1/deg\left(\alpha \right)}$, the infimum being over all non-zero non-cyclotomic $\alpha$ lying with its conjugates outside $V$. We evaluate $L\left(V\right)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L\left(V\right)$, which we call the transfinite Lehmer constant ${L}_{\infty }\left(V\right)$.Also, we prove the converse to Langevin’s Theorem, which states that $L\left(V\right)>1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for ${L}_{\infty }\left(V\right)$.

## How to cite

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Dubickas, Artūras, and Smyth, Chris J.. "The Lehmer constants of an annulus." Journal de théorie des nombres de Bordeaux 13.2 (2001): 413-420. <http://eudml.org/doc/248715>.

@article{Dubickas2001,
abstract = {Let $M (\alpha )$ be the Mahler measure of an algebraic number $\alpha$, and $V$ be an open subset of $\mathbb \{C\}$. Then its Lehmer constant$L (V)$ is inf $M (\alpha )^\{1/ \deg (\alpha )\}$, the infimum being over all non-zero non-cyclotomic $\alpha$ lying with its conjugates outside $V$. We evaluate $L(V)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L (V)$, which we call the transfinite Lehmer constant $L_\infty (V)$.Also, we prove the converse to Langevin’s Theorem, which states that $L (V) &gt; 1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for $L_\infty (V)$.},
author = {Dubickas, Artūras, Smyth, Chris J.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Lehmer constant; Mahler measure; polynomial zeros},
language = {eng},
number = {2},
pages = {413-420},
publisher = {Université Bordeaux I},
title = {The Lehmer constants of an annulus},
url = {http://eudml.org/doc/248715},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Dubickas, Artūras
AU - Smyth, Chris J.
TI - The Lehmer constants of an annulus
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 413
EP - 420
AB - Let $M (\alpha )$ be the Mahler measure of an algebraic number $\alpha$, and $V$ be an open subset of $\mathbb {C}$. Then its Lehmer constant$L (V)$ is inf $M (\alpha )^{1/ \deg (\alpha )}$, the infimum being over all non-zero non-cyclotomic $\alpha$ lying with its conjugates outside $V$. We evaluate $L(V)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L (V)$, which we call the transfinite Lehmer constant $L_\infty (V)$.Also, we prove the converse to Langevin’s Theorem, which states that $L (V) &gt; 1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for $L_\infty (V)$.
LA - eng
KW - Lehmer constant; Mahler measure; polynomial zeros
UR - http://eudml.org/doc/248715
ER -

## References

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7. [Le1] D.H. Lehmer, Factorisation of certain cyclotomic functions. Ann. of Math. (2) 34 (1933), 461-479. Zbl0007.19904MR1503118JFM59.0933.03
8. [Le2] D.H. Lehmer, Review of [La3]. Math. Rev.89j:11025.
9. [M] M. Mignotte, Sur un théorème de M. Langevin. Acta Arith.54 (1989), 81-86. Zbl0641.12003MR1024420
10. [RS] G. Rhin, C.J. Smyth, On the absolute Mahler measure of polynomials having all zeros in a sector. Math. Comp.65 (1995), 295-304. Zbl0820.11064MR1257579
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12. [Sm] C.J. Smyth, On the measure of totally real algebraic integers. J. Austral. Math. Soc. Ser. A30 (1980), 137-149. Zbl0457.12001MR607924

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