<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>n ≥</italic> 2 be an integer and denote by <italic>θn</italic> the real root in (0, 1) of the trinomial <italic>Gn</italic>(<italic>X</italic>) = −1 + <italic>X</italic> + <italic>Xn</italic>. The sequence of Perron numbers <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msub><mrow><mo>(</mo><msubsup><mi>θ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></msub></math><tex-math>$(\theta _n^{ - 1} )_{n \ge 2} $</tex-math></alternatives></inline-formula> tends to 1. We prove that the Conjecture of Lehmer is true for <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mo>{</mo><msubsup><mi>θ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>|</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>}</mo></math><tex-math>$\{ \theta _n^{ - 1} |n \ge 2\} $</tex-math></alternatives></inline-formula> by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots <italic>θn</italic>, <italic>zj,n</italic>, of <italic>Gn</italic>(<italic>X</italic>) lying in <italic>|z|</italic> &lt; 1, as a function of <italic>n, j</italic> only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mtext>M</mtext><mo>(</mo><msub><mi>G</mi><mi>n</mi></msub><mo>)</mo><mo>=</mo><mtext>M</mtext><mo>(</mo><msub><mi>θ</mi><mi>n</mi></msub><mo>)</mo><mo>=</mo><mtext>M</mtext><mo>(</mo><msubsup><mi>θ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></math><tex-math>${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$</tex-math></alternatives></inline-formula> of the trinomials <italic>Gn</italic> as a function of <italic>n</italic> only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mo>{</mo><msubsup><mi>θ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>|</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>}</mo></math><tex-math>$\{ \theta _n^{ - 1} |n \ge 2\} $</tex-math></alternatives></inline-formula>, with a minoration of the house <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0006_eq_005.png"></inline-graphic>, and a minoration of the Mahler measure M(<italic>Gn</italic>) better than Dobrowolski’s one. The angular regularity of the roots of <italic>Gn</italic>, near the unit circle, and limit equidistribution of the conjugates, for <italic>n</italic> tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.</abstract>

Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers (θn−1)n≥2$(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1. We prove that the Conjecture of Lehmer is true for {θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $ by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn, zj,n, of Gn(X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures M(Gn)=M(θn)=M(θn−1)${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$ of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for {θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house , and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots of Gn, near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

LAMA, CNRS UMR 5127, Université Savoie Mont Blanc, 73376 - Le Bourget-du-Lac,

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers (θn−1)n≥2$(\theta _n^{ -...

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

Published Online: Jan 13, 2017

Page range: 79 - 139

Received: Mar 20, 2015

Accepted: Sep 14, 2015

DOI: https://doi.org/10.1515/udt-2016-0006

Keywords
Mahler measure, trinomial, Lehmer Conjecture, asymptotic expansion, divergent series, Perron number, Pisot number, Schinzel-Zassenhaus conjecture, Smyth conjecture, Lind-Boyd Conjecture, Boyd Conjecture, Erdős-Turán-Mignotte-Amoroso, discrepancy function, limit equidistribution

© 2016 Jean-Louis Verger-Gaugry, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

Published Online: Jan 13, 2017

Page range: 79 - 139

Received: Mar 20, 2015

Accepted: Sep 14, 2015

DOI: https://doi.org/10.1515/udt-2016-0006

KeywordsMahler measure, trinomial, Lehmer Conjecture, asymptotic expansion, divergent series, Perron number, Pisot number, Schinzel-Zassenhaus conjecture, Smyth conjecture, Lind-Boyd Conjecture, Boyd Conjecture, Erdős-Turán-Mignotte-Amoroso, discrepancy function, limit equidistribution

© 2016 Jean-Louis Verger-Gaugry, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Mahler measure, trinomial, Lehmer Conjecture, asymptotic expansion, divergent series, Perron number, Pisot number, Schinzel-Zassenhaus conjecture, Smyth conjecture, Lind-Boyd Conjecture, Boyd Conjecture, Erdős-Turán-Mignotte-Amoroso, discrepancy function, limit equidistribution