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Autre Publication Scientifique Année : 2022

An attack of the Conjecture of Lehmer by the dynamical zeta function of the β -shift, and the modulo p problem

Résumé

The present work proposes an attack of the Conjecture of Lehmer by the dynamical zeta function of the β -shift to prove that this Conjecture is true (math NT> arXiv:1911.10590 (29 Oct 2021)). In 1933 Lehmer asked the question about the existence of integer polynomials having a Mahler measure different of one, smaller than Lehmer’s number (and arbitrarily close to one). The problem of Lehmer became a Conjecture, stating that there exists a universal lower bound > 1 to the Mahler measures of the nonzero algebraic integers which are not roots of unity. The problem of the minoration of the Mahler measure of algebraic integers is a very deep one and has been extended in the theory of heights in arithmetic geometry. The main ingredients arise from the lenticular poles of the dynamical zeta functions ζ_β (z) of the R ́enyi–Parry arithmetical dynamical (“β -shift”), with β > 1 any real number tending to one, to which a lenticular measure can be associated, satisfying a Dobrowolski-type inequality with the dynamical degree of β . When β runs over the set of nonzero reciprocal algebraic integers, under some assumptions, the lenticular poles are identified with conjugates of β , using Kala-V ́avra’s periodic representation theorem (2019), and this lenticular measure is identified with a minorant of the Mahler measure of β . Though expressed as hypergeometric functions (Mellin, 1915) the lenticularity of the poles only appears when using their Poincar ́e asymptotic expansions, in the angular sector guessed by M. Langevin, G. Rhin and C. Smyth, G. Rhin and Q. Wu. We show that the search for very small Mahler measures calls for investigating the factorization of integer polynomials in a class of lacunary polynomials canonically associated to the functions ζ β (z), that this problem is linked to the number of zeroes of these polynomial in F p , to their asymptotic limit when p tends to infinity, and questions on the existence of modular forms by the Langlands program. Whether Lehmer’s number is the smallest Mahler measure > 1 of algebraic integers re- mains open. References: D. Dutykh, J.-L. Verger-Gaugry, Alphabets, rewriting trails, periodic representation in algebraic basis, Res. Number Theory 7:64 (2021). J.-L. Verger-Gaugry, On a class of lacunary almost-Newman polynomials modulo p and density theorems, Unif. Distrib. Theory 17, No 1 (2022), 29–54. J.-L. Verger-Gaugry, A Dobrowolski-type inequality for the poles of the dynamical zeta function of the beta-shift, submitted (2022), J.-L. Verger-Gaugry, An universal minoration of the Mahler measure of real reciprocal algebraic integers, (2022).
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hal-03704316 , version 1 (01-07-2022)

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  • HAL Id : hal-03704316 , version 1

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Jean-Louis Verger-Gaugry. An attack of the Conjecture of Lehmer by the dynamical zeta function of the β -shift, and the modulo p problem. 2022. ⟨hal-03704316⟩
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