On limit dynamical zeta functions and Mahler measure of height one trinomials

Jean-Louis Verger-Gaugry (1, 2)

1 LAMA - Laboratoire de Mathématiques
2 IF - Institut Fourier

Let χ_3 (m) be the Jacobi symbol (m/3) (χ_3 (m) is equal to 0, 1, −1 according to χ (m) whether m ≡ 0, 1 or 2 (mod 3)). Let L(s, χ_3 (m)) = ∑_{m≥1} χ_3 (m) m^(-s) be the Dirichlet L-series. Denote by Λ = 1.38135 . . . = exp(3 √ 3 /(4π) L(2, χ_3 (m))). The Mahler measure of a polynomial P is denoted by M(P). Due to the invariance properties of M , the following conjecture covers all cases of height one trinomials. Conjecture (Smyth). For all integers n ≥ 4, k ≥ 1 such that gcd(n, k) = 1, k < n/2, • M(z^n + z^k + 1) < Λ if and only if 3 divides n + k, • M(z^n − z^k + 1) < Λ with n odd if and only if 3 does not divide n + k, • M(z^n − z^k − 1) < Λ with n even if and only if 3 does not divide n + k. and lim M(±1 ± X^k + X^n ) = Λ. n→∞ Flammang (2014) proved this conjecture for large n. Verger-Gaugry (2016) proved it for the family (−1+z+z n ) n≥4 using Poincaré asymptotic expansions, introduced on purpose in Number Theory from the N body problem in celestial mechanics and the Theory of perturbations of Poincar ́e (1895–1905), and using Smyth’s and Boyd’s techniques dedicated to trinomials. In the present study, the links between the dynamical zeta functions ζ_β (z) of the R ́enyi–Parry dynamical systems (β -shift) and the minoration of Mahler measures in such limit problems are investigated, in particular the problem of Lehmer. With this new approach, we will present the examples given by these trinomials, in particular the new type of minoration of the Mahler measure, better than Dobrowolski’s and Voutier’s ones for the dominant roots (Perron numbers).

Domaines: Théorie des nombres [math.NT]

Jean-Louis Verger-Gaugry. On limit dynamical zeta functions and Mahler measure of height one trinomials. Zeta Functions 6 - Moscou 2016, D. Fiorilli, F. Jouve, P. Lebacque, A. Zykin, Dec 2016, Moscou, Russia. ⟨hal-03757546⟩