VG IV.– Asymptotic Expansions of the Mahlermeasures

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let θn be the unique root of the trinomial Gn(z) :=−1 + z + zn in (0,1).

θ−1n −1 + z + zn

θ−1n < β < θ

−1n−1 −1 + x + xn + xm1 + xm2 + . . .+ xms + . . .

where m1−n ≥ n−1, mq+1−mq ≥ n−1

θ−1n−1 −1 + z + zn−1

covers all cases of reciprocal algebraic integers β tending to 1.Lehmer’s Problem : find a minorant > 1 of M(β ).

(θ−1n ) tends to 1. β tends to 1 equivalently n tends to infinity.

n =: dynamical degree of β , denoted dyg(β ).

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Recall the geometry of the roots

a)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

b)

-1 -0.5 0.5 1

-1

-0.5

0.5

1

FIGURE: Dynamical degree 37 : a) the 37 zeroes of G37(x) =−1 + x + x37,b) the zeroes of fβ (x) = G37(x) + . . .+. The lenticulus of roots of fβ is a slightdeformation of the restriction of the lenticulus of roots of G37 to the angularsector |argz|< π/18, off the unit circle. The other roots (nonlenticular) of fcan be found in a narrow annular neighbourhood of |z|= 1.

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θ−1n < β < θ

−1n−1 −1 + x + xn + xm1 + xm2 + . . .+ xms + . . .

where s ≥ 1, m1−n ≥ n−1, mq+1−mq ≥ n−1 for 1≤ q < s

denoted =: fβ (x), is the inverse of the dynamical zeta function ζβ (z)(up to the sign) of the β -shift, equivalently is the generalized Fredholmdeterminant of the Perron-Frobenius operator associated with theβ -transformation. Called Parry Upper function at β .

its zeroes := eigenvalues of the transfer operators, := poles of ζβ (z).

2 collections of zeroes : close to |z|= 1, off |z|= 1 (lenticular).

The lenticular roots are identified as Galois conjugates of 1/β , that isof β .

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2 relations, with uniform convergence on every K in Ωn, domain strictlyincluded in D(0,1) containing only the lenticular roots, avoiding the others

• existence of lenticuli (each sN in the class B)

limN→∞

sN(z) = fβ (z)

• rewriting polynomials (in base β ), allowing the identification :

lims→∞

As(z)Pβ (z) = fβ (z)

Hence, since fβ (z) has no zero in Ωn except the lenticular zeroes,every zero of Pβ (z) is a zero of fβ (z).Conversely, since 1/β is a simple zero of Pβ (z) and of fβ (z), thatlenticuli are limits of the lenticuli of the irreducible factors of the sN(z),then the complete lenticulus of fβ (z) is a lenticulus of zeroes ofPβ (z).

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Method :

1) With a good theory of approximation of the lenticular roots, obtainan asymptotic expansion of M(θ

−1n ) from the lenticuli of roots in

|arg(z)|< π/3, with n tending to ∞,

2) extended to any θ−1n < β < θ

−1n−1, with reduced lenticuli of roots, in

|arg(z)|< π/18, and n = dyg(β ) tending to ∞.

good theory := Poincare asymptotic expansions (as in the N-bodyProblem in Celestial Mechanics).

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- the geometry of the lenticular roots should be as exact as possible :Galois theory is valid up to degree 5. After, for higher degrees, for thetrinomials −1 + x + xn, we use the Poincare asymptotic expansions ofthe roots of the trinomials −1+x +xn, together with Rouche’s method,

- limit equidistribution for the other roots (angular distribution).

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minorant :

Mr (β ) :=: Mlenticulus(β ) := ∏i lenticular min1, |β (i)|−1

expressed as an asymptotic expansion of dyg(β ).

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Lehmer’s Conjecture : solved for (θ−1n ) by using (i) the dynamical

zeta functions ζθ−1n

(z) of the θ−1n -shift, and (ii) their poles and their

Poincare asymptotic expansions.

JLVG : On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials andAsymptotic Expansions, Uniform Distribution Theory J. 11 (2016), 79–139.

Galois theory : exact expressions of roots as functions of thecoefficients, up to degree 5. Here not exact, but extension using thetheory of divergent series developped by H. Poincare in CelestialMechanics.

Les Methodes Nouvelles de la Mecanique Celeste, 1892, 1899 -1905,Lecons de Mecanique Celeste, 1905

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Let us put θn = 1− tn with 0 < t < n. Then

tn

= (1− tn

)n. (1)

Let us show that t < Logn. Let g(x) = xex be the increasing function ofthe variable x on R. This implies t

n = (1− tn )n < e−t ⇔ g(t) < n. Since

n < nLogn for n ≥ 3 and g(Logn) = nLogn, we deduce the claim.Taking the logarithm we obtain

Log t−Logn = nLog(1− tn

) = −t− 12

t2

n− 1

3t3

n2 − . . . .

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The identity

t + Log t +12

t2

n+

13

t3

n2 + . . . = Logn (2)

has now to be inversed in order to obtain t as a function of n. For doingthis, we put t = Logn + w . Equation (2) transforms into the followingequation in w :

(Logn + w) + Log(Logn + w) +12

(Logn + w)2

n+

13

(Logn + w)3

n2 + . . . = Logn.

We deduce

w + Log(Logn) + Log(1 +w

Logn) +

12

Log2nn

(1 +

wLogn

)2

+13

Log3

n2

(1 +

wLogn

)3+ . . . = 0.

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Since

nLog(1− Lognn

) + Logn =−12

Log2nn− 1

3Log3n

n2 − . . .

and that

Log(1 +w

Logn) =

wLogn

− w2

2Log2n+

w3

3Log3n− . . .

we have :

LogLogn−nLog(1− Lognn

)−Logn

= w[−1− 1

Logn− Logn

n− Log2n

n2 − Log3nn3 − . . .

]

+ w2[

12Log2n

− 12n− Logn

n2 − 64

Log2nn3 − . . .

]+ . . . . (3)

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The coefficient coeff(w) of w is

−1− 1Logn

− Lognn

(1 +

Lognn

+

(Logn

n

)2

+ . . .

)=−nLogn−n + Logn(Logn)(n−Logn)

.

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We deduce

w =(Logn)(n−Logn)(LogLogn−nLog(1− Logn

n )−Logn)

−nLogn−n + Logn+ . . . , (4)

which gives the expression of D(θn).Let us write w in (4) as D(w) + u, where u denotes the remaindingterms. Putting w = D(w) + u in (3) we obtain, for large n,

0 = u coeff(w) + D(w)2 12Log2n

+ . . . .

Since, for large n, coeff(w)∼=−1 and D(w)∼=−LogLogn, we deduce :

u ∼= O

((LogLogn

Logn

)2),

with a constant 1/2 involved in O ( ). We deduce the tail tl(θn) of θn.

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Prop. :Let n ≥ 2. The root θn can be expressed as : θn = D(θn) + tl(θn) withD(θn) =

1− Lognn

(1−

(n−Logn

nLogn + n−Logn

)(LogLogn−nLog

(1− Logn

n

)−Logn

))(5)

and

tl(θn) =1n

O

((LogLogn

Logn

)2), (6)

with the constant 1/2 involved in O ( ).

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Theorem (Smyth, VG ’16)

Let χ3 be the uniquely specified odd character of conductor 3(χ3(m) = 0,1 or −1 according to whether m ≡ 0, 1 or 2 (mod 3),equivalently χ3(m) =

(m3

)the Jacobi symbol), and denote

L(s,χ3) = ∑m≥1χ3(m)

ms the Dirichlet L-series for the character χ3. Then

limn→+∞

M(Gn) = limn→+∞

M(θ−1n ) = exp

(3√

34π

L(2,χ3)

)

= exp(−1π

∫π/3

0Log

(2 sin

(x2

))dx)

= 1.38135 . . . =: Λ.

Λ > 1, then L Cj.

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Theorem (VG ’16)

Let n0 be an integer such that π

3 > 2πLogn0

n0, and let n ≥ n0. Then

M(Gn) =

(lim

m→+∞M(Gm)

)(1 + r(n)

1Logn

+ O(

LogLognLogn

)2)

with the constant 1/6 involved in the Big O, and with r(n) real,|r(n)| ≤ 1/6.

Take n0 = 18. Improvment of Dobrowolski’s inequality :

Corollary (VG ’16)

M(θ−1n ) > Λ− Λ

6

(1

Logn

), n ≥ n1 = 2.

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Sch-Z for (θ−1n ) :

Theorem (VG ’16)For all n ≥ 2,

θ−1n = θ

−1n ≥ 1 +

cn,

with c = 2(θ−12 −1) = 1.2360 . . . reached only for n = 2, and,

θ−1n = θ

−1n > 1 +

(Logn)(

1− LogLognLogn

)n

.

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Given a point α ∈Q×, of degree d = [Q(α) : Q], we define the unitBorel measure (probability)

µα =1d ∑

σ

δσ(α)

on C×, the sum being taken over all d embeddings σ : Q(α)→ C. Asequence αk of points in Q× is said to be strict if any properalgebraic subgroup of Q× contains αk for only finitely many values of k .

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Theorem (VG ’16)

Let θ−1n | n = 2,3,4, . . . be the infinite sequence of Perron numbers in

C× which are the dominant roots of the trinomials G∗n. Then

µθ−1n→ µT, n→+∞, weakly, (7)

or equivalently,µθn → µT, n→+∞, weakly, (8)

i.e., for all bounded, continuous functions f : C×→ C,∫fdµ

θ−1n→∫

fdµT, n→+∞.

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roots close to |z|= 1. Limit equidistribution of conjugates towards thenormalized Haar measure on |z|= 1, as n→ ∞.

-1 -0.5 0.5 1

-1

-0.5

0.5

1

FIGURE: Roots of G∗12(z), n = 12.

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-1 -0.5 0.5 1

-1

-0.5

0.5

1

FIGURE: Roots of G28(z), n = 28.

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-1 -0.5 0.5 1

-1

-0.5

0.5

1

FIGURE: Roots of G∗50(z), n = 50.

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-1 -0.5 0.5 1

-1

-0.5

0.5

1

FIGURE: Roots of G∗100(z), n = 100.

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-1 -0.5 0.5 1

-1

-0.5

0.5

1

FIGURE: Roots of G∗150(z), n = 150.

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-1 -0.5 0.5 1

-1

-0.5

0.5

1

FIGURE: Roots of G∗200(z), n = 200.

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Let κ := 0.171573 . . . be the value of the maximum of the function

a→ κ(1,a) :=1−exp(−π

a )

2exp( πa )−1 on (0,∞). Let S := 2arcsin(κ/2) = 0.171784 . . ..

Denote

Λr µr := exp(−1

π

∫ S

0Log

[1 + 2sin( x2 )−

√1−12sin( x

2 ) + 4(sin( x2 ))2

4

]dx)

= 1.15411 . . . , a value slightly below Lehmer’s number 1.17628 . . .

Theorem

limdyg(β)→∞

∏ω∈Lβ

|ω|−1 = Λr µr .

It is the limit lenticular contribution of the Parry Upper function.

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Further : identify the lenticuli of zeroes of the Parry Upper function at β withlenticuli of conjugates of β , so that

limdyg(β )→∞

∏ω∈Lβ

|ω|−1 = limdyg(β )→∞

Mr (β ) = Λr µr .

Further : the asymptotic expansions of the roots of fβ (z) lying in β

gives an asymptotic expansion of the lenticular minorant of the Mahlermeasure :

Theorem (Dobrowolski type minoration)

Let β be a nonzero algebraic integer which is not a root of unity suchthat dyg(β )≥ 260. Then

M(α)≥ Λr µr −Λr µrS2π

( 1Log(dyg(β ))

)28 / 31

JLVG, On the Conjecture of Lehmer, Limit Mahler Measure ofTrinomials and Asymptotic Expansions, Uniform Distribution Theory J.11 (2016), 79–139.

JLVG (Sept. 2017) : A Proof of the Conjecture of Lehmer and of theConjecture of Schinzel-Zassenhaus

arXiv.org > math > arXiv :1709.03771

version v2 (2018).

D. Dutykh and JLVG, On the Reducibility and the Lenticular Sets ofZeroes of Almost Newman Polynomials Having Lacunarity Controlled aMinima, preprint (2018).

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Lehmer’s Conjecture :

Theorem (VG ’17)For any nonzero algebraic integer α which is not a root of unity,

M(α)≥ θ−1259 = 1.016126 . . .

Schinzel Zassenhaus’s Conjecture :

Theorem (VG ’17)Schinzel-Zassenhaus’s conjecture is true. Let α be a nonzeroalgebraic integer which is not a root of unity. Then

α ≥ 1 +C

deg(α)with C = θ

−1259−1 = 0.016126 . . . .

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Pf. : Let α 6= 0 be an algebraic integer which is not a root of unity. SinceM(α) = M(α−1) there are three cases to be considered :(i) the house of α satisfies α ≥ θ

−15 ,

(ii) the dynamical degree of α satisfies : 6≤ dyg(α) < 260,(iii) the dynamical degree of α satisfies : dyg(α)≥ 260.

In case (i), M(α)≥ θ−15 ≥ θ

−1259 and the claim holds true.

In the second case, since M(α) is the product of α by the moduli ofthe conjugates of modulus > 1, we have M(α)≥ α , thereforeM(α)≥ θ

−1259.

In case (iii), the Dobrowolski type inequality gives the following lowerbound of the Mahler measure

M(α)≥Λr µr −Λr µr arcsin(κ/2)

π Log(dyg(α))≥Λr µr −

Λr µr arcsin(κ/2)

π Log(259),= 1.14843 . . .

This lower bound is numerically greater than θ−1259 = 1.016126 . . ..

Therefore, in any case, the lower bound θ−1259 of M(α) holds true. We

deduce the general minorant on M.31 / 31