DIRICHLET SERIES FOR FINITE COMBINATORIAL RANK DYNAMICS

G. EVEREST, R. MILES, S. STEVENS, AND T. WARD

Draft May 24, 2007

Abstract. We introduce a class of group endomorphisms – those of finite combinatorial rank – exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.

Contents

1. Introduction 1 2. Finite combinatorial rank 4 3. Dynamical Dirichlet series 10 4. Examples 13 5. Proof of Theorems 3.3 and 3.4 16 6. Proof of Theorem 3.5 20 7. Proof of Theorem 3.6 22 References 35

1. Introduction

A closed orbit τ of length |τ| = n for a map α: X → X is a set of the form

{x, α(x), α^{2}(x), …, α^{n}(x) = x}

with cardinality n (in our setting X will always be a compact metric space and α a continuous map). Dynamical analogues of the prime

2000 Mathematics Subject Classification. 37C30; 26E30; 12J25. This research was supported by E.P.S.R.C. grant EP/C015754/1.

number theorem concern the asymptotic behavior of quantities like

π_{α}(N) = |{τ a closed orbit of α: |τ| ⩽ N}| .

When X has a metric structure with respect to which α is hyperbolic, results of Parry and Pollicott [11], [12], [13] and others apply to give a precise understanding of the growth properties of π_{α}. Without hyperbolicity less is known: Waddington [16] considered the case of a quasihyperbolic toral automorphism, and the authors [3] found asymptotics for connected S-integer systems with S finite (these are isometric extensions of hyperbolic automorphisms of arithmetic origin).

Writing

F_{α}(n) = |{x ∈ X: α^{n}x = x}|

for the number of points fixed by α^{n}, the dynamical zeta function of α is defined by

ζ_{α}(z) = exp ∑_{n=1}^{∞} \frac{z^{n}}{n} F_{α}(n)

which has a formal expansion as an Euler product,

ζ_{α}(z) = ∏_{τ} (1 − z^{|τ|})^{−1} ,

where the product is taken over all closed orbits of α. In the hyperbolic and quasihyperbolic settings, the dynamical zeta function is either rational or has a meromorphic extension beyond its radius of convergence, allowing analytic methods to be used to obtain asymptotic estimates for π_{α}. For S-integer systems with S finite the dynamical zeta function typically has a natural boundary, so more direct methods have to be used in [3]. In all these cases the asymptotic takes the form

π_{α}(N) ∼ \frac{e^{h(α)N}}{N} Ψ(N),

where Ψ is an explicit almost-periodic function bounded away from zero and infinity, constant in the hyperbolic cases, and h(α) is the topological entropy of α. The S-integer systems with S finite may be viewed as arithmetical perturbations of (quasi)hyperbolic systems, and the orbit growth is still essentially exponential.

Our purpose here is to describe orbit growth for a class of dynamical systems much further from hyperbolicity: those of finite combinatorial rank. The dynamical system (X, α) has finite combinatorial rank if the sequence of periodic point counts (F_{α}(n)) has finite rank in a canonical partially ordered set of integer sequences (see Definition 2.9). These systems are found to possess an orbit structure which allows Dirichlet series to be used directly. The simplest non-trivial example of a group automorphism with finite combinatorial rank is the automorphism α dual to the map x ↦ 2x on the localization ℤ_{(3)} of the integers at the prime 3 (the periodic point data for this map is the same as that of the map z ↦ z^{2} on {z ∈ ℂ: z^{3^{k}} = 1 for some k ⩾ 0}). By [2], this system has F_{α}(n) = |2^{n} − 1|_{3}^{−1}, and Stangoe [14] used this to show the asymptotic

π_{α}(N) = \frac{1}{log 3} log(N) + O(1),

for all N ⩾ 1.

The main results concern an algebraic system (X, α) of finite combinatorial rank, and the number O_{α}(n) of closed orbits of length n under α. We associate the Dirichlet series

d_{α}(z) = ∑_{n=1}^{∞} \frac{O_{α}(n)}{n^{z}}

to (X, α) and use this to study the orbit-growth function

π_{α}(N) = ∑_{n⩽N} O_{α}(n).

In Theorem 3.3 we find an exact expression for the Dirichlet series and use this to show that, in the connected case, d_{α}(z) is a rational function of the variables {c^{−z}: c ∈ C} where C is a finite set of positive integers (see Example 4.1 for a simple instance of this rationality result).

The asymptotic behaviour of π_{α} is governed by the abscissa of convergence σ of d_{α} and the order K of the pole at σ. For the case σ = 0, Theorem 3.6(1) shows that

π_{α}(N) = C (log N)^{K} + O ((log N)^{K−1}) .

Example 7.2 illustrates this result. For σ > 0 the situation is more involved. In combinatorial rank one (in which case the pole at σ is necessarily simple), Theorem 3.5 shows that

π_{α}(N) = δ(N)N^{σ} + O(1)

where δ is an explicit oscillatory function bounded away from zero and infinity. An instance of this phenomena may be found in Example 4.3.

For the general case of a simple pole, we show in Theorem 3.6(2) a Chebychev result of the form

AN^{σ} ⩽ π_{α}(N) ⩽ BN^{σ}

for constants A, B > 0. Given the oscillatory function that arises in combinatorial rank one, no stronger asymptotic can be expected.

Finally, and surprisingly, in the case σ > 0 and K ⩾ 2, Theorem 3.6(3) gives the exact asymptotic

π_{α}(N) ∼ CN^{σ} (log N)^{K−1} .

This comes about because the higher-order pole introduces a factor which randomizes the oscillatory behaviour seen in the case of a simple pole. This is most easily seen in Example 7.4.

Typically, particularly in analytic number theory, the route from the singular behaviour of a Dirichlet series to a counting function goes via a Tauberian theorem. However, the presence of singularities of d_{α} arbitrarily close together along the line ℜ(z) = σ hampers this approach. Indeed, the oscillatory term in Theorem 3.5 is incompatible with the conclusion one might hope to reach via Tauberian methods. In simpler situations like Example 4.1 (where the singularites on the line ℜ(z) = σ are equally spaced), Agmon’s Tauberian theorem [1] or Perron’s Theorem [6, Th. 13] can be applied. However, it is far easier to obtain asymptotics directly from the series expression for d_{α}.

The letter C is used to denote various constants independent of variables n, N, x and so on; ℕ_{0} denotes ℕ ∪ {0}.

2. Finite combinatorial rank

Let X be a compact metrizable abelian group of finite topological dimension and let α be a continuous ergodic epimorphism of X with finite topological entropy. Let D denote the class of such dynamical systems (X, α). For (X, α) ∈ D, Lemma 2.1 below will show that F_{α}(n) is finite for all n ⩾ 1.

Consider the ring ℤ^{ℕ} with the product ordering ≼ (so (a_{n}) ≼ (b_{n}) if a_{n} ⩽ b_{n} for all n ∈ ℤ). For a subset A ⊂ ℤ^{ℕ}, a ∈ A is said to have finite rank in A if there is some C > 0 such that all strictly decreasing chains of elements of A starting at a have length at most C.

Pontryagin duality gives a one-to-one correspondence between elements of D and certain countable modules over the domain R = ℤ[t] as follows. The additive group M = X̂ has the structure of an Rmodule by identifying x ↦ tx with the map α̂ and extending in an obvious way to polynomials. The module M satisfies the following conditions:

(1) M is countable (since X is metrizable);

(2) the map x ↦ tx is a monomorphism of M (since α is onto);

(3) the set of associated primes Ass(M) is finite and consists entirely of non-zero principal ideals, none of which are generated by cyclotomic polynomials (since X is finite-dimensional, α is ergodic and h(α) < ∞);

(4) for each 𝔭 ∈ Ass(M),

m(𝔭) = dim_{𝕂(𝔭)} M_{𝔭} < ∞,

where 𝕂(𝔭) denotes the field of fractions of R/𝔭 (since X is finite-dimensional and h(α) < ∞).

Conversely, if M is an R-module satisfying these four properties and α_{M} denotes the epimorphism of X_{M} = M̂ dual to x ↦ tx on M, then

(X_{M}, α_{M}) ∈ D.

Denote the class of all such R-modules M by D̂.

The next lemma gives a formula for the number of periodic points of an element of D in terms of valuations of sequences of the form

θ(𝔭) = (\overline{t}^{n} − 1)_{n⩾1},

where \overline{t} denotes the image of t in R/𝔭 for some prime ideal 𝔭 associated to M. In order to describe this, suppose that 𝔭 ⊂ R is a principal prime ideal so 𝕂(𝔭) is a global field with finite residue class fields. Let P(𝔭), P_{0}(𝔭), P_{∞}(𝔭) denote the places, finite places and infinite places of 𝕂(𝔭) respectively. In what follows, if v denotes a normalized additive valuation corresponding to a place in P_{0}(𝔭), | · |_{v} is defined by | · |_{v} = |𝔎_{v}|^{−v(·)}, where 𝔎_{v} is the residue class field. For ease of notation, if P is any subset of the set of places of a global field K then write

|x|_{P} = ∏_{v∈P} |x|_{v},

for x ∈ K, with the convention that |x|_{∅} = 1. Finally, note that each v ∈ P_{0}(𝔭) also induces a map | · |_{v}: 𝕂(𝔭)^{ℕ} → ℚ^{ℕ}, and |θ(𝔭)|_{v} is a unit in ℚ^{ℕ} provided 𝔭 is not generated by a cyclotomic polynomial.

Write F_{α} for the sequence (F_{α}(n))_{n⩾1}, and for sequences a = (a_{n}) and b = (b_{n}) write ab, a^{k} for the sequences (a_{n}b_{n}), (a_{n}^{k}) respectively.

Lemma 2.1. Let M ∈ D̂. Then there exist 𝔭_{1}, …, 𝔭_{r} ∈ Ass(M) and sets of places P_{1}, …, P_{r} with P_{i} ⊂ P_{0}(𝔭_{i}), 1 ⩽ i ⩽ r, such that

F_{α_{M}} = ∏_{i=1}^{r} |θ(𝔭_{i})|_{P_{i}}^{−1}.  (1)

Proof. This is shown in [8, Th. 1.1]. □

For M ∈ D̂, set

[M] = {F_{α_{L}}: L is a submodule of M} ⊂ ℤ^{ℕ}.

Lemma 2.1 and the next result together show how [M] relates to the simpler constituent sets of sequences of the form [𝕂(𝔭)]. The members of a set like [𝕂(𝔭)] are sequences of periodic points for S-integer systems. Write 2^{P} for the set of subsets of P.

Lemma 2.2. Let 𝔭 ⊂ R be a principal prime ideal which is not generated by a cyclotomic polynomial, and set

P = {v ∈ P_{0}(𝔭): |R/𝔭|_{v} is bounded}.

Then [𝕂(𝔭)] is the image of the map ψ: 2^{P} → ℤ^{ℕ} where

ψ(Q) = |θ(𝔭)|_{Q}^{−1}.

Proof. The proof of [8, Th. 1.1] shows that [𝕂(𝔭)] ⊂ ψ(2^{P}), so it remains to show the reverse inclusion. Fix a subset Q ⊂ P. If Q = ∅, set L = 𝕂(𝔭), otherwise set L = ⋂_{v∈Q} R_{v}, where R_{v} is the discrete valuation ring of 𝕂(𝔭) corresponding to v. Then L is a submodule of 𝕂(𝔭) with F_{α_{L}} = ψ(Q) by construction. □

It follows that

[M] ⊂ ∏_{𝔭∈Ass(M)}[𝕂(𝔭)]^{m(𝔭)}  (2)

and each [𝕂(𝔭)] has a description as in Lemma 2.2.

Proposition 2.3. For any R-module M ∈ D̂ the following holds. (1) For each 𝔭 ∈ Ass(M), the set [𝕂(𝔭)] contains a greatest element

s(𝔭) = F_{α_{R/𝔭}} .

Furthermore, for any a ∈ [𝕂(𝔭)], a^{−1}s(𝔭) ∈ [𝕂(𝔭)].

(2) The set [M] contains a greatest element

F_{α_{L}} = ∏_{𝔭∈Ass(M)}s(𝔭)^{m(𝔭)},

given by a Noetherian submodule L ⊂ M.

Proof. (1) This follows from the description given by Lemma 2.2: The sequence ψ(P) = F_{α_{R/𝔭}} is the greatest element of [𝕂(𝔭)] and for any subset Q ⊂ P, we have ψ(Q)ψ(P \ Q) = ψ(P).

(2) Let

s = ∏_{𝔭∈Ass(M)}s(𝔭)^{m(𝔭)},

then (1) and Lemma 2.1 show that F_{α_{M}} ≼ s. The result follows by observing that there is a Noetherian submodule L ⊂ M with a prime filtration in which each of the primes 𝔭 ∈ Ass(M) appears with multiplicity m(𝔭) and no other primes appear. It follows that F_{α_{L}} = s. □

Example 2.4. If α_{M} is a quasihyperbolic toral epimorphism then F_{α_{M}} is the greatest element of [M]. More generally, for any Noetherian module M ∈ D̂, F_{α_{M}} is the greatest element of [M].

The next result gives a convenient characterization of the finite-rank sequences in a poset of the form [𝕂(𝔭)]; this will be useful when considering dynamical systems of finite combinatorial rank in the next section.

Theorem 2.5. Let 𝔭 ⊂ R be a principal prime ideal which is not generated by a cyclotomic polynomial and let a ∈ [𝕂(𝔭)]. Then a has finite rank if and only if there is a finite set of places Q ⊂ P_{0}(𝔭) such that |θ(𝔭)|_{Q}^{−1} = a.

The proof of Theorem 2.5 requires the following lemma, taken from [9, Lem. 4.9]. Recall that 𝔎_{v} denotes the residue class field of a place v, write ℓ_{v} for the multiplicative order of the image of t in 𝔎_{v}^{×} and r_{v} for the (absolute) residue degree.

Lemma 2.6. Let 𝔭 ⊂ R be a principal prime ideal which is not generated by a cyclotomic polynomial. Let v ∈ P_{0}(𝔭) have the property that |\overline{t}|_{v} = 1 and let p = char(𝔎_{v}). Then there exist constants D ⩾ 1, E ⩾ 0 such that

|θ(𝔭)_{n}|_{v}^{−1} = \{ \begin{matrix}1 & if ℓ_{v} ∤ n, \\ Dp^{r_{v} ord_{p}(n)} & if ℓ_{v}|n, char(𝕂(𝔭)) = 0 and ord_{p}(n) ⩾ E, \\ |θ(𝔭)_{ℓ_{v}}|_{v}^{−|n|_{p}^{−1}} & if ℓ_{v}|n and char(𝕂(𝔭)) > 0. \end{matrix}

Proof of Theorem 2.5. Let 𝕂 = 𝕂(𝔭), θ = θ(𝔭) and let P, ψ be as in Lemma 2.2. For any Q ⊂ P, set

Q̃ = {v ∈ Q: |θ|_{v}^{−1} ≠ 1}.

Notice that if |\overline{t}|_{v} < 1 then |θ|_{v} = 1, so |\overline{t}|_{v} = 1 for all v ∈ P̃.

Suppose that a has finite combinatorial rank and that Q ⊂ P satisfies ψ(Q) = a. Assume Q̃ is infinite. By choosing a strictly decreasing chain

Q̃ = Q_{1} ⊃ Q_{2} ⊃ ⋯,

it follows that

a = ψ(Q_{1}) ≻ ψ(Q_{2}) ≻ ⋯,

contradicting the assumption that a has finite combinatorial rank. Hence Q̃ is finite and satisfies ψ(Q̃) = a.

Conversely, suppose there is a finite set Q ⊂ P with ψ(Q) = a. Without loss of generality, assume that Q = Q̃. If there are only finitely many distinct Q′ ⊂ P̃ with ψ(Q′) ≼ a, the required result follows. Hence, for a contradiction assume there are infinitely many such Q′.

First suppose that char(𝕂) = 0. Then the set of rational primes

S = {char(𝔎_{v}): v ∈ Q}

is finite. Furthermore, using Lemma 2.6, there exist positive constants c, D, E such that for each n ⩾ 1 with min_{p∈S}{ord_{p}(n)} ⩾ E,

a_{n} = |θ_{n}|_{Q}^{−1} ⩽ Dc^{j(n)},  (3)

where j(n) = max{ord_{p}(n): p ∈ S}.

There are only finitely many v ∈ P_{0}(𝕂) with char(𝔎_{v}) equal to a given prime. So, since there are infinitely many distinct Q′ ⊂ P̃ with ψ(Q′) ≼ a, it is possible to choose Q′ to contain a place v with char(𝔎_{v}) ∉ S. For r ⩾ 0, set

n(r) = ℓ_{v}q^{r} ∏_{p∈S} p^{E},

where q = char(𝔎_{v}). Since j(n(r)) is bounded, (3) shows that a_{n(r)} is also bounded. However, again applying Lemma 2.6, for a sufficiently large choice of r,

|θ_{n(r)}|_{Q′}^{−1} ⩾ |θ_{n(r)}|_{v}^{−1} > a_{n(r)},

contradicting ψ(Q′) ≼ a.

Now assume that char(𝕂) = p > 0. Then R/𝔭 = 𝔽_{p}[\overline{t}] and \overline{t} is transcendental over 𝔽_{p}. Since Q is finite, Lemma 2.6 shows that the set {a_{n}: ord_{p}(n) = 0, n ⩾ 1} is bounded. There are, by assumption, infinitely many distinct Q′ ⊂ P̃ with ψ(Q′) ≼ a so it is possible to choose such a Q′ to contain a place v for which |𝔎_{v}| is larger than this bound. However, |θ_{ℓ_{v}}|_{v} ⩽ |𝔎_{v}|^{−1} and since ord_{p}(ℓ_{v}) = 0,

|θ_{ℓ_{v}}|_{Q′}^{−1} ⩾ |θ_{ℓ_{v}}|_{v}^{−1} ⩾ |𝔎_{v}| > a_{ℓ_{v}},

contradicting ψ(Q′) ≼ a again. □

Example 2.7. The poset [𝕂(t−2)], which contains the sequences of periodic point counts arising from algebraic factors of the map x ↦ 2x on the solenoid ℚ̂, is illustrated in Figure 1 (the figure shows a small part of the full poset: the first level from the bottom has infinitely many sequences, parameterised by the rational primes). The greatest element corresponds to the circle doubling map x ↦ 2x on 𝕋 = ℝ/ℤ, and the central sequence represents ψ(Q), where Q is the set of Mersenne primes. According to Theorem 2.5, ψ(Q) has finite rank if and only

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023
[figure]
1, 3, 7, 3, 31, 63, 127, 51, 511, 1023 1, 3, 7, 15, 31, 63, 127, 255, 511, 93
[figure]
1, 3, 7, 3, 31, 63, 127, 51, 511, 93
⋮
1, 3, 7, 3, 31, 63, 127, 3, 7, 93
⋮
1, 3, 7, 3, 1, 63, 1, 3, 7, 3
[figure]
1, 3, 1, 3, 1, 9, 1, 3, 1, 3 1, 1, 7, 1, 1, 7, 1, 1, 7, 1
[figure]
1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Figure 1. Part of the Hasse diagram for [𝕂(t − 2)].

if there are finitely many Mersenne primes. The least element corresponds to the map x ↦ 2x on ℚ̂, corresponding to ψ(Q) where Q is the set of all rational primes.

Remark 2.8. The S-integer systems studied by Chothi, Everest and Ward [2] give an alternative way to describe the periodic point sequences we study. In the connected case, they may be described as follows. Fix an algebraic number field 𝕂 with set of places P(𝕂) and set of infinite places P_{∞}(𝕂), an element of infinite multiplicative order ξ ∈ 𝕂^{∗}, and a set S ⊂ P(𝕂)\P_{∞}(𝕂) with the property that |ξ|_{w} ⩽ 1 for all w ∉ S ∪ P_{∞}(𝕂). The associated ring of S-integers is

R_{S} = {x ∈ 𝕂: |x|_{w} ⩽ 1 for all w ∉ S ∪ P_{∞}(𝕂)}.

Let X be the character group of R_{S}, and define an endomorphism α to be the dual of the map x ↦ ξx on R_{S}. By [2, Lemma 5.2] the number of points in X fixed by α^{n} is

F_{α}(n) = ∏_{w∈S∪P_{∞}(𝕂)}|ξ^{n} − 1|_{w}.

In the language of [2], the systems we are interested in are the co-finite ones (those for which P(𝕂) \ S is a finite set) and finite products of them. Using the product formula for global fields, the periodic point formula when S is co-finite reduces to one of the factors in (1).

In view of the canonical inclusion (2) and the results of this section, we make the following definition.

Definition 2.9. Let M ∈ D̂. Then α_{M} has finite combinatorial rank if F_{α_{M}} has finite rank in

∏_{𝔭∈Ass(M)}[𝕂(𝔭)]^{m(𝔭)}.

For the rest of the paper, we assume that (X, α) ∈ D.

3. Dynamical Dirichlet series

Writing O_{α}(n) for the number of orbits of length n under α, we have the following arithmetic relation between periodic points and orbits,

F_{α}(n) = ∑_{d|n} d O_{α}(d).

By Möbius inversion,

O_{α}(n) = \frac{1}{n} ∑_{d|n} μ(n/d) F_{α}(d).  (4)

Definition 3.1. The dynamical Dirichlet series associated to the map α is the formal series

d_{α}(z) = ∑_{n=1}^{∞} \frac{O_{α}(n)}{n^{z}}.

Example 3.2. The examples we wish to study are group epimorphisms of finite combinatorial rank, but the quadratic map α : x ↦ 1 − cx^{2} on the interval [−1, 1] at the Feigenbaum value c = 1.401155 ⋯ (see Feigenbaum’s lecture notes [5]) gives a particularly simple example of a dynamical Dirichlet series. This map has

O_{α}(n) = \begin{cases}1 & if n = 2^{k} for some k ⩾ 0; \\ 0 & if not, \end{cases}

so d(z) = \frac{1}{1−2^{−z}}. In this example it is clear that π_{α}(N) = \frac{log N}{log 2} + O(1) in accordance with Theorem 3.6, but it is important to emphasise that in general the asymptotic growth statements cannot be deduced from the analytic behaviour of the Dirichlet series alone.

Using convolution of Dirichlet series (see [15, Sec. 3.7]), an alternative way of expressing the relation (4) is

d_{α}(z) = \frac{1}{ζ(z + 1)} ∑_{n=1}^{∞} \frac{F_{α}(n)/n}{n^{z}},  (5)

where as usual

ζ(z + 1) = ∑_{n=1}^{∞} \frac{1/n}{n^{z}}.

For systems of finite combinatorial rank, this observation is extremely useful: it is possible to extract ζ(z + 1) as a factor of the series on the right-hand side of (5), resulting in an exact expression for orbit counting. Note that only combinatorial properties of the zeta function are of importance here.

Recall that the set of rational primes p with

p | F_{α}(n) for some n ∈ ℕ  (6)

is finite when F_{α} has finite rank.

Theorem 3.3. Let (X, α) be a system of finite combinatorial rank and let Q be the set of rational primes given by (6). Then d_{α}(z) is a linear combination of Dirichlet series of the form

∑_{e∈ℕ_{0}^{P}} \frac{φ_{P} (Λ(e))}{(bφ_{P} (e))^{z}} ,

where b ∈ ℕ, P ⊂ Q, Λ : ℕ_{0}^{P} → ℕ_{0}^{P} and φ_{P} (e) = ∏_{p∈P} p^{e_{p}}.

Furthermore, all these quantities can be determined explicitly. In particular, denoting the local field corresponding to a place v by 𝕂_{v} and the residue degree of v by r_{v},

Λ(e)_{p} = −e_{p} + ∑_{v∈S(p)} r_{v}λ_{v}(e_{p}),

where S(p) is a set of places with residue class fields of characteristic p, and

λ_{v}(e_{p}) = \{ \begin{matrix}e_{p} & if char(𝕂_{v}) = 0, \\ η_{v}p^{e_{p}} & if char(𝕂_{v}) > 0, \end{matrix}

for some positive integer constant η_{v}.

Theorem 3.3 and its proof allow us to deduce the following result for connected systems.

Theorem 3.4. Let (X, α) be a connected system of finite combinatorial rank. Then there is a finite set C ⊂ ℕ with the property that d_{α}(z) is a rational function of the variables {c^{−z}: c ∈ C}.

In principle, Theorem 3.3 provides an exact formula for orbit counting, since it implies that π_{α}(N) is a linear combination of expressions of the form

∑_{e∈ℕ_{0}^{P} ,} ; φ_{P} (e)⩽N/bφ_{P} (Λ(e)).

However, a closed asymptotic expression involving elementary functions is often more desirable, the prime number theorem being a case in point. Our focus is on connected systems, although Theorem 3.3 may also be applied in the disconnected case (cf. Example 4.4).

Theorem 3.5. Let (X, α) be a connected system of combinatorial rank one and let r_{v} be the residue degree of the associated place. Then

π_{α}(N) = \{ \begin{matrix}C log N + O(1) & if r_{v} = 1, \\ δ(N)N^{r_{v}−1} + O(1) & otherwise, \end{matrix}

where δ(N) is an explicit oscillatory function bounded away from zero and infinity.

Theorem 3.5 shows that an elementary asymptotic formula for π_{α} cannot be expected in general. Typically, an oscillatory function similar to that in Theorem 3.5 appears when the abscissa of convergence of d_{α} is greater than zero and is a simple pole; we provide a Chebychev result in this case. In all other cases, an elementary asymptotic formula is found.

Theorem 3.6. Let (X, α) be a connected system of finite combinatorial rank whose Dirichlet series d_{α} has abscissa of convergence σ ⩾ 0. Let K denote the order of this pole.

(1) If σ = 0 then there is a constant C > 0 with

π_{α}(N) = C (log N)^{K} + O ((log N)^{K−1}) .

(2) If σ > 0 and K = 1 then there are constants A, B, N_{0} > 0 such that for all N > N_{0},

AN^{σ} ⩽ π_{α}(N) ⩽ BN^{σ}.

(3) If σ > 0 and K ⩾ 2 then there is a constant C > 0 with

π_{α}(N) ∼ CN^{σ} (log N)^{K−1} .

Remark 3.7. (1) If (X, α) is a connected system of finite combinatorial rank and X has topological dimension one, the proof of Theorem 3.3 shows that d_{α} has abscissa of convergence σ = 0. Thus, the exact asymptotic of Theorem 3.6(1) applies. In general, there is no straightforward relationship between d_{α} and the topological dimension of X.

(2) Using Hlawka’s bounds [7] on equidistribution in terms of discrepancy and Baker’s theorem, the asymptotic in Theorem 3.6(3) could potentially be improved to give an error term.

4. Examples

One of the characteristic features of non-trivial S-integer dynamical systems is the extremely complex behavior of their dynamical zeta functions (see [3], [4], [17] for example), so the rationality of the Dirichlet series for systems of finite combinatorial rank is a little surprising.

Example 4.1. The basic reason for rationality of the Dirichlet series is already visible in the simplest non-trivial example given by the map α dual to the map x ↦ 2x on ℤ_{(3)} = ℤ[\frac{1}{p}: p a prime ≠ 3]. We have

F_{α}(n) = |2^{n} − 1|_{3}^{−1}

and hence

F_{α}(n) = \begin{cases}3^{ord_{3}(n)+1} & if n is even; \\ 1 & otherwise. \end{cases}

From (5) we know that

d_{α}(z) = ζ(z + 1)^{−1} (∑_{2∤n} \frac{F_{α}(n)}{n^{z+1}} + ∑_{2|n} \frac{F_{α}(n)}{n^{z+1}} ) .

Treating each term separately,

∑_{2∤n} \frac{F_{α}(n)}{n^{z+1}} = ∑_{n⩾1} \frac{1}{n^{z+1}} − ∑_{n⩾1} \frac{1}{(2n)^{z+1}} = ζ(z + 1) (1 − \frac{1}{2^{z+1}} )

and (on writing an even n as 2 · k · 3^{e} with 3 ∤ k)

∑_{2|n} \frac{F_{α}(n)}{n^{z+1}} = ∑_{e⩾0} ∑_{k⩾1,3∤k} \frac{3^{e+1}}{(2k3^{e})^{z+1}}

= \frac{3}{2^{z+1}} ∑_{e⩾0} \frac{1}{3^{ez}} ∑_{k⩾1,3∤k} \frac{1}{k^{z+1}}

= \frac{3}{2^{z+1}} · \frac{1}{1 − 3^{−z}} · ζ(z + 1) (1 − \frac{1}{3^{z+1}} ) .

Thus,

d_{α}(z) = 1 − \frac{1}{2^{z+1}} (1 − \frac{3}{1 − 3^{−z}} (1 − \frac{1}{3^{z+1}} )) = 1 + \frac{1}{2^{z}} (\frac{1}{1 − 3^{−z}} ) .

Note that the abscissa of convergence is σ = 0 and this is a simple pole.

The orbit-growth function π_{α} may be obtained by extracting the coefficients from the series expression for d_{α},

π_{α}(N) = 1 + ∑_{e⩾0, 2·3^{e}⩽N} 1 = 1 + ∑_{e=0}^{⌊log_{3}(N/2)⌋} 1 = \frac{log N}{log 3} + O(1).

Notice that in Example 4.1 the rational expression for d_{α} has infinitely many singularities along the line ℜ(z) = 0 at the points 2kπi/ log 3 for k ∈ ℤ. A less direct method for obtaining an asymptotic estimate for π_{α}(N) is provided by Agmon’s Tauberian theorem [1] which applies in situations like this, but as the next example shows, the general case has even worse analytic properties. It also illustrates the impact of additional places.

Example 4.2. Let α be the map dual to x ↦ 2x on ℤ_{(3)} ∩ ℤ_{(5)}. Then

F_{α}(n) = |2^{n} − 1|_{3}^{−1}|2^{n} − 1|_{5}^{−1},

so

F_{α}(n) = \begin{cases}1 & if n is odd; \\ 3 · 3^{e_{1}} & if n = 2 · k · 3^{e_{1}}, 2 ∤ k, 3 ∤ k; \\ 15 · 3^{e_{1}} · 5^{e_{2}} & if n = 4 · k · 3^{e_{1}} · 5^{e_{2}}, 3 ∤ k, 5 ∤ k. \end{cases}

A similar calculation to that used in Example 4.1 shows that

d_{α}(z) = 1 − \frac{1}{2^{z+1}} + \frac{3}{2^{z+1}} (1 − \frac{1}{3^{z+1}} − \frac{1}{2^{z+1}} + \frac{1}{6^{z+1}} ) \frac{1}{1 − 3^{−z}}

+\frac{15}{4^{z+1}} (1 − \frac{1}{3^{z+1}} − \frac{1}{5^{z+1}} + \frac{1}{15^{z+1}} ) \frac{1}{(1 − 3^{−z})(1 − 5^{−z})}.

Here, the abscissa of convergence σ = 0 is a double pole. An asymptotic expression for π_{α} is obtained in Section 7.1.

Notice that in Example 4.2 not only are there infinitely many singularities with ℜ(z) = 0, but there are singularities that are arbitrarily close together.

The next example illustrates the situation when a higher topological dimension is allowed. This may also be regarded as allowing places with a higher residue degree.

Example 4.3. Let 𝕂 = ℚ(\sqrt[3]{5}), with ring of integers

𝔒 = ℤ + \sqrt[3]{5}ℤ + (\sqrt[3]{5})^{2}ℤ = ℤ[\sqrt[3]{5}]

as ℤ-modules. Notice that 𝔒 ≅ R/(t^{3} − 5) via the map

\sqrt[3]{5} ↦ \overline{t} = t + (t^{3} − 5).

There are two primes lying above the prime 2 of ℤ, namely (2, 1 + \overline{t}) and 𝔪 = (2, 1 + \overline{t} + \overline{t}^{2}). The place v corresponding to 𝔪 has residue degree 2, since 𝔒/𝔪 ≅ 𝔽_{4}.

Let X = 𝔒̂_{𝔪} and let α be the map dual to x ↦ \sqrt[3]{5} x. Then X has topological dimension 3 and

F_{α}(n) = |\overline{t}^{n} − 1|_{v} = \begin{cases}1 & 3 ∤ n; \\ 16 · 2^{2 ord_{2}(n)} & otherwise. \end{cases}

Using a similar method to the previous examples, the dynamical Dirichlet series for this example is,

d_{α}(z) = 1 + \frac{5}{3^{z}} + \frac{8}{6^{z}} (\frac{1}{1 − 2^{1−z}} ) .

In this case, the abscissa of convergence is σ = 1 and this is a simple pole. Using a similar method as before (or by using (4) directly), we obtain the closed formula

O_{α}(n) = \begin{cases}1 & n = 1; \\ 5 & n = 3; \\ 4 · 2^{e} & n = 3 · 2^{e} and e ⩾ 1; \\ 0 & otherwise. \end{cases}

It follows that

π_{α}(N) = 6 + 4 ∑_{e=1}^{⌊log_{2}(N/3)⌋} 2^{e}

= \frac{8}{3} 2^{−\{log_{2}(N/3)\}}N − 2,  (7)

for all N ⩾ 6, where {·} denotes the fractional part of a real number. Notice that (7) includes the oscillatory factor δ(N) appearing in Theorem 3.5. In this example,

lim sup_{N→∞} δ(N) = \frac{8}{3}

and

lim inf_{N→∞} δ(N) = \frac{4}{3}.

To conclude this section, the following example shows that the situation is quite different for disconnected systems. Here there is no rational expression for d_{α} although the series organizes orbit data in a convenient way, giving an exact expression for π_{α}.

Example 4.4. Let M be the discrete valuation ring 𝔽_{3}[t]_{(t−1)} obtained by localizing the domain 𝔽_{3}[t] at the prime ideal (t − 1). Since the additive group of M is torsion, X = M̂ is zero-dimensional. Furthermore, the map α dual to x ↦ tx has periodic points counted by a single place v of 𝔽_{3}(t) corresponding to M. That is,

F_{α}(n) = |t^{n} − 1|_{(t−1)}^{−1} = 3^{|n|_{3}^{−1}}.

Therefore,

d_{α}(z) = ζ(z + 1)^{−1} ∑_{n⩾1} \frac{3^{|n|_{3}^{−1}}}{n^{z+1}}

= ζ(z + 1)^{−1} ∑_{e⩾0} \frac{3^{3^{e}}}{(3^{e})^{z+1}} ∑_{k⩾1,3∤ k} \frac{1}{k^{z+1}}

= (1 − \frac{1}{3^{z+1}} ) ∑_{e⩾0} \frac{3^{3^{e}}/ 3^{e}}{(3^{e})^{z}.}

Hence, we have the following exact expression for π_{α},

π_{α}(N) = ∑_{e⩾0, 3^{e}⩽N} \frac{3^{3^{e}}}{3^{e}} − \frac{1}{3} ∑_{e⩾0, 3^{e}⩽N/3} \frac{3^{3^{e}}}{3^{e}}

= 1 + \frac{2}{3} ∑_{e⩾0, 3^{e}⩽N} \frac{3^{3^{e}}}{3^{e}} .

A further calculation shows that

lim sup_{N→∞} \frac{Nπ_{α}(N)}{3^{N}} = \frac{2}{3}

and

lim inf_{N→∞} \frac{Nπ_{α}(N)}{3^{N}} = 0.

5. Proof of Theorems 3.3 and 3.4

The general case requires an inclusion–exclusion argument to deal with the congruence conditions arising from multiple terms in (1) and multiple places in each set P_{i}.

Since (X, α) has finite combinatorial rank, the sets P_{i} appearing in (1) are all finite. Let T be the union of the sets P_{i}, and let ξ_{v} for v ∈ T be the image of t in the appropriate field 𝕂(𝔭). Notice that each ξ_{v} has infinite multiplicative order and we may assume |ξ_{v}|_{v} = 1. For any finite non-empty set S ⊂ T write

f_{S}(n) = ∏_{v∈S} |ξ_{v}^{n} − 1|_{v}^{−1}.  (8)

Thus f_{T} (n) = F_{α}(n). Let ℓ_{v} be the multiplicative order of the image of ξ_{v} in the residue field 𝔎_{v} of v. For S ⊂ T and a set Q of rational primes, define

S(Q) = {v ∈ S: char(𝔎_{v}) = p for some p ∈ Q} ⊂ S.

If Q = {p}, we also write S(p) for S(Q). Consider

d_{α}(z) = ζ(z + 1)^{−1} ∑_{n=1}^{∞} \frac{f_{T} (n)}{n^{z+1}}.

Let ℓ_{S} = lcm{ℓ_{v}: v ∈ S}. Then the collection {N_{S}, S ⊂ T }, defined by

N_{S} = {n ∈ ℕ: ℓ_{S}|n and ℓ_{v} ∤ n forall v ∈ T \ S},

forms a partition of ℕ. Hence,

d_{α}(z) = ζ(z + 1)^{−1} ∑_{S⊂T} ∑_{n∈N_{S}} \frac{f_{S}(n)}{n^{z+1}},  (9)

since f_{S} = f_{T} on N_{S} by Lemma 2.6.

Let E = ℕ_{0}^{Q}, where

Q = Q(S) = {char(𝔎_{v}): v ∈ S}.

For any P ⊂ Q, we extend the definition of φ_{P} given in the statement of Theorem 3.3 to accomodate vectors e = (e_{p}) ∈ E by setting

φ_{P} (e) = ∏_{p∈P} p^{e_{p}}.

Each integer n ∈ N_{S} can be written in the form

n = ℓ_{S}kφ_{Q}(e),  (10)

where k ∈ ℕ satisfies

p ∤ k for all p ∈ Q  (11)

and

ℓ_{v} ∤ ℓ_{S}kφ_{Q}(e) for all v ∈ T \ S.  (12)

It is more convenient to rephrase (12) according to the following partition of E. Let

e_{0} = max{ord_{p}(ℓ_{v}): p ∈ Q, v ∈ T \ S}  (13)

and let e_{0} ∈ E be the vector with each entry equal to e_{0}. For P ⊂ Q, set

E_{P} = {e ∈ E: e_{p} ⩾ e_{0} for all p ∈ P, e_{p} < e_{0} for all p ∈ Q \ P },

so the sets {E_{P} : P ⊂ Q} partition E. For each q in the finite set φ_{Q\P} (E_{P}), let

E_{P} (q) = {e ∈ E_{P} : φ_{Q\P} (e) = q}.

Then, for e ∈ E_{P} (q),

ℓ_{S}φ_{Q}(e) = ℓ_{S}φ_{P} (e)q,

and if v ∈ T \ S,

gcd(ℓ_{v}, ℓ_{S}φ_{Q}(e)) = gcd(ℓ_{v}, ℓ_{S}φ_{P} (e)q) = gcd(ℓ_{v}, ℓ_{S}φ_{P} (e_{0})q).

Hence, provided e ∈ E_{P} (q), condition (12) becomes,

j ∤ k for all j ∈ {ℓ_{v}/ gcd(ℓ_{v}, ℓ_{S}φ_{P} (e_{0})q): v ∈ T \ S}.  (14)

For n of the form (10), by Lemma 2.6, f_{S}(n) = f_{S}(ℓ_{S}φ_{Q}(e)). Furthermore, using the partition of E given above, the inner sum in (9) may be written

∑_{P ⊂Q} ∑_{q∈φ_{Q\P} (E_{P})} ∑_{e∈E_{P} (q)}(ℓ_{S}φ_{Q}(e))^{−z−1}f_{S}(ℓ_{S}φ_{Q}(e)) ∑_{k} k^{−z−1}, (15)

where k runs through all natural numbers satisfying (11) and (14). Using inclusion-exclusion, the inner sum in (15) is of the form

ζ(z + 1)g(c^{−z}, c ∈ C),  (16)

where C ⊂ ℕ is a finite set of constants and g is a multivariate polynomial with rational coefficients, the monomial terms of which have total degree one. Using (15) and (16), by cancelling the zeta function with its reciprocal and adjusting C and g to absorb ℓ_{S}^{−z−1}, (9) becomes

d_{α}(z) = ∑_{S⊂T} ∑_{P ⊂Q} ∑_{q∈φ_{Q\P} (E_{P})} d_{S,P,q}(z),

where

d_{S,P,q}(z) = g(c^{−z}, c ∈ C) ∑_{e∈E_{P} (q)} φ_{Q}(e)^{−z−1}f_{S}(ℓ_{S}φ_{Q}(e)). (17)

Again using Lemma 2.6, for each p ∈ Q, there exists e′_{p} such that for all v ∈ S(p) and all e ∈ E with e_{p} ⩾ e′_{p},

f_{v}(ℓ_{S}φ_{Q}(e)) = c_{v}p^{r_{v}ω_{v}(e_{p})},  (18)

where c_{v} is a positive integer constant, ω_{v}(e_{p}) = e_{p} if the local field 𝕂_{v} corresponding to v has zero characteristic and ω_{v}(e_{p}) = η′_{v}p^{e_{p}} for some constant η′_{v} ∈ ℕ, otherwise. Without loss of generality, the steps already carried out may be performed with e_{0} replaced by the maximum of the value given by (13) and max{e′_{p}: p ∈ Q}.

For any e ∈ E_{P} (q), φ_{Q}(e) = φ_{P} (e)q, so the summand in (17) is

(φ_{P} (e)q)^{−z−1}f_{S(Q\P)}(ℓ_{S}q)f_{S(P)}(ℓ_{S}φ_{P} (e)).  (19)

Therefore,

d_{S,P,q}(z) = g(c^{−z}, c ∈ C) ∑_{e∈E_{P} (q)} φ_{P} (e)^{−z−1}f_{S(P)}(ℓ_{S}φ_{P} (e)),

where g and C have been adjusted to accommodate the factor

q^{−z−1}f_{S(Q\P)}(ℓ_{S}q),

appearing in (19). Using (18), it follows that

d_{S,P,q}(z) = C × g(c^{−z}, c ∈ C) ∑_{e∈E_{P} (q)} \frac{φ_{P} (Ω(e) − e)}{φ_{P} (e)^{z}}  (20)

where C > 0 and Ω : ℕ_{0}^{P} → ℕ_{0}^{P} is given by,

Ω(e)_{p} = ∑_{v∈S(p)} r_{v}ω_{v}(e_{p}).

By the definition of E_{P} (q) and φ_{P}, the sum in (20) is

∑_{e∈ℕ_{0}^{P} ,} ; e_{p}⩾e_{0}, p∈P \frac{φ_{P} (Ω(e) − e)}{φ_{P} (e)^{z}} = ∑_{e∈ℕ_{0}^{P}} \frac{φ_{P} (Ω(e + e_{0}) − e − e_{0})}{φ_{P} (e + e_{0})^{z}}  (21)

where e_{0} ∈ ℕ_{0}^{P} is the vector with each entry equal to e_{0}. For each v ∈ S(p) with char(𝕂_{v}) = p, set η_{v} = p^{e_{0}}η′_{v} and let Λ : ℕ_{0}^{P} → ℕ_{0}^{P} be defined in terms of η_{v} as in the statement of Theorem 3.3. Then

Ω(e + e_{0}) − e = Λ(e) + d,

where d = (d_{p}) ∈ ℕ_{0}^{P} is given by

d_{p} = e_{0} ∑_{v∈S(p),} ; char(𝕂_{v})=0r_{v}.

Therefore, by (20) and (21),

d_{S,P,q}(z) = g(c^{−z}, c ∈ C) ∑_{e∈ℕ_{0}^{P}} \frac{φ_{P} (Λ(e))}{φ_{P} (e)^{z}},  (22)

where g has been modified to absorb the factor Cφ_{P} (d)φ_{P} (e_{0})^{−1−z}. This concludes the proof of Theorem 3.3.

To prove Theorem 3.4, first note that since X is assumed to be connected, all places arise from number fields. Setting,

r_{S,p} = ∑_{v∈S(p)} r_{v},

it follows from the definition of Λ that

d_{S,P,q}(z) = g(c^{−z}, c ∈ C) ∑_{e∈ℕ_{0}^{P}} ∏_{p∈P} p^{e_{p}(r_{S,p}−1−z)}.

Therefore, the sum contains a geometric series corresponding to each prime p ∈ P. Hence

d_{S,P,q}(z) = g(c^{−z}, c ∈ C) ∏_{p∈P}(1 − p^{r_{S,p}−1−z})^{−1}.  (23)

Since d_{α}(z) is a finite sum of expressions of this form, this proves the theorem.

For the proof of the asymptotic results we will need a specific description of the function g appearing in (22) in order to control cancellation between terms of different signs in various counting arguments. Notice that for a finite set of positive integers J with 1 ∉ J,

∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}} > 0,

where c_{I} = lcm{i: i ∈ I}.

Lemma 5.1. Suppose g ≠ 0. Then

g(c^{−z}, c ∈ C) = C × ∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}} b_{I}^{−z},

where C > 0, J is a finite set of positive integers with 1 ∉ J and b_{I} = Bc_{I} for each I ⊂ J, for some fixed constant B ∈ ℕ.

This may be seen by going through the steps in the proof above.

6. Proof of Theorem 3.5

By Lemma 2.6, we have constants ℓ_{v}, a_{1}, …, a_{d}, D and a prime p with

F_{α}(n) = \begin{cases}1 & ℓ_{v} ∤ n; \\ a_{e} & n = ℓ_{v}p^{e}k, where p ∤ k, 1 ⩽ e ⩽ d; \\ Dp^{r_{v}e} & n = ℓ_{v}p^{e}k, p ∤ k, e > d. \end{cases}

By the method used in Section 5, we deduce that

d_{α}(z) = 1 − ℓ_{v}^{−z−1} + ∑_{e=1}^{d} a_{e} (ℓ_{v}p^{e})^{−z−1} + \frac{cb^{−z}}{1 − p^{m−z}} (1 − \frac{1}{p^{z+1}} )

where b = ℓ_{v}p^{d+1}, c = \frac{D}{ℓ_{v}} p^{(d+1)(r_{v}−1)} and m = r_{v} − 1. It follows that – up to an error uniformly bounded in N – we can compute π_{α}(N) by considering the dominant term

\frac{b^{−z}}{1 − p^{m−z}} (1 − \frac{1}{p^{z+1}} ) = \frac{b^{−z}}{1 − p^{m−z}} − \frac{(bp)^{−z}}{p(1 − p^{m−z})}.  (24)

The first term in (24) contributes

∑_{e: bp^{e}⩽N} p^{me} = ∑_{e=0}^{⌊log_{p}(N/b)⌋} p^{me}  (25)

and the second term in (24) contributes

\frac{1}{p} ∑_{e: bp^{e+1}⩽N} p^{me} = \frac{1}{p} ∑_{e=1}^{⌊log_{p}(N/b)⌋} p^{m(e−1)}

= p^{−m−1} ( ∑_{e=0}^{⌊log_{p}(N/b)⌋} p^{me}) − p^{−m−1}. (26)

The total contribution from (25) and (26) is therefore

(1 − p^{−m−1}) ∑_{e=0}^{⌊log_{p}(N/b)⌋} p^{me} + O(1).  (27)

If r_{v} = 1, then m = 0 and (27) becomes,

(1 − \frac{1}{p}) log_{p}(N/b) + O(1) = \frac{p − 1}{p log p} log N + O(1),

If r_{v} > 1, then m > 0 and (27) becomes,

C ∑_{e=0}^{⌊log_{p}(N/b)⌋} p^{me} + O(1) = C · \frac{p^{m(1+⌊log_{p}(N/b)⌋)} − 1}{p^{m} − 1} + O(1)
= δ(N)N^{m} + O(1)

for some constant C and a function δ satisfying the requirements of Theorem 3.5.

7. Proof of Theorem 3.6

Since X is connected, d_{α} has a rational expression which is a finite sum of terms of the form (23). By Lemma 5.1, each such term may be written as

F (z) = C × G(z) ∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}} b_{I}^{−z},

where

G(z) = ∏_{p∈P}(1 − p^{n_{p}−z})^{−1}

for some n = (n_{p}) ∈ ℕ_{0}^{P}. Let

m = max{n_{p} : p ∈ P } and L = |{p ∈ P: n_{p} = m}| .

Note that the Dirichlet series for G has abscissa of convergence m and G has a pole of order L at z = m.

By showing that the coefficient of (z − m)^{−L} in the Laurent series for F is positive, it will follow that for at least one such F, m = σ, L = K and that there can be no F for which m > σ nor any F for which m = σ and L > K.

The coefficient of (z − m)^{−L} in the Laurent series for G about m is κ = ∏_{p∈P} κ_{p} where

κ_{p} = \begin{cases}(log p)^{−1} & if n_{p} = m \\ (1 − p^{n_{p}−m})^{−1} & if n_{p} < m \end{cases}

Hence, the coefficient of (z − m)^{−L} in the Laurent series for F is

Cκ ∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}b_{I}^{m}} = \frac{Cκ}{B^{m}} ∑_{I⊂J} \frac{(−1)^{|I|}}{(lcm\{i : i ∈ I\})^{m+1}}

= \frac{Cκ}{B^{m}} ∑_{I⊂J} \frac{(−1)^{|I|}}{(lcm\{i^{m+1} : i ∈ I\})}

= \frac{Cκ}{B^{m}} ∑_{I⊂J′} \frac{(−1)^{|I|}}{(lcm\{i : i ∈ I\})},

where J′ = {j^{m+1} : j ∈ J}. Since the constants κ, B, C and the result of the sum are all positive, so too is the coefficient of (z − m)^{−L} in the Laurent series for F.

Now consider the contribution to π_{α} arising from F. For any m ∈ ℤ^{P} and x ⩾ 0, set

S_{x}^{P} (m) = ∑_{e∈ℕ_{0}^{P} ,} ; φ_{P} (e)⩽xφ_{P} (me),

where integer vectors are multiplied term-by-term. Extracting the coefficients from the Dirichlet series for F, it follows that F contributes

C × ∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}} S_{N/b_{I}}^{P}(n)  (28)

to π_{α}(N). This expression will be the main tool for obtaining our asymptotics.

The following result gives a Chebychev estimate for the individual terms S_{N/b_{I}}^{P}(n).

Lemma 7.1. Let m = (m_{p}) ∈ ℤ^{P} and x ⩾ 0. Then there exist constants B ⩾ A > 0 and x_{0} ⩾ 0 such that for all x > x_{0},

Ax^{m}(log x)^{j} ⩽ S_{x}^{P} (m) ⩽ Bx^{m}(log x)^{j},

where m = max{0, m_{p} : p ∈ P } and

j = \begin{cases}|\{p ∈ P: m_{p} = m\}| & if m = 0, \\ |\{p ∈ P: m_{p} = m\}| − 1 & if m > 0. \end{cases}

Proof. First suppose m = 0. For any e ∈ ℕ_{0}^{P},

e_{p} ⩽ |P |^{−1} log_{p} x for all p ∈ P ⇒ φ_{P} (e) ⩽ x.

Therefore,

S_{x}^{P} (m) ⩾ ∏_{p∈P} ∑_{e_{p}=0}^{⌊|P |^{−1} log_{p} x⌋} p^{m_{p}e_{p}}
⩾ A(log x)^{j},

for some positive constant A. On the other hand, for any e ∈ ℕ_{0}^{P},

φ_{P} (e) ⩽ x ⇒ e_{p} ⩽ log_{p} x for all p ∈ P.

So,

S_{x}^{P} (m) ⩽ ∏_{p∈P} ∑_{e_{p}=0}^{⌊log_{p} x⌋} p^{m_{p}e_{p}}
⩽ B(log x)^{j},

for some positive constant B ⩾ A.

Now suppose m > 0. The proof is by induction on |P |. If |P | = 1 then the result is obvious, so assume |P | > 1. Choose q ∈ P such that m_{q} = m, set P′ = P \ {q} and m′ = (m_{p})_{p∈P′}. Write

S_{x}^{P} (m) = ∑_{e∈ℕ_{0}^{P′},} ; φ_{P′}(e)⩽xφ_{P′}(m′e) ∑_{e_{q}=0}^{⌊log_{q}(x/φ_{P′}(e))⌋} q^{me_{q}}.  (29)

The inner sum in (29) is a geometric series with sum

\frac{q^{m(1+⌊log_{q} x/φ_{P′}(e)⌋)} − 1}{q^{m} − 1}.

This expression is bounded above by

C_{1}x^{m}φ_{P′}(−me) − C_{2}

for some constants C_{1}, C_{2} > 0. It follows from (29) and the m = 0 case that

S_{x}^{P} (m) ⩽ C_{1}x^{m}S_{x}^{P′}(m′ − m1) − C_{2}S_{x}^{P′}(m′)
⩽ C_{3}x^{m}(log x)^{j} − C_{2}S_{x}^{P′}(m′),

for some constant C_{3} > 0. Upon noting that either (1) max{m′_{p} : p ∈ P′} < m or

(2) max{m′_{p} : p ∈ P′} = m and |{p ∈ P′: m′_{p} = m}| − 1 < j, this completes the inductive step. A similar argument also gives the lower bound. □

7.1. Proof of Theorem 3.6(1). In this case all expressions of the form (28) contributing to π_{α}(N) have n = 0. Example 4.2 illustrates some of the issues that arise in this setting when more than one prime is involved.

Example 7.2. (Example 4.2 revisited) Let α be the map dual to x ↦ 2x on ℤ_{(3)} ∩ ℤ_{(5)}, so

F_{α}(n) = |2^{n} − 1|_{3}^{−1}|2^{n} − 1|_{5}^{−1}.

The term in d_{α} that determines the asymptotic growth in π_{α} is

15 (\frac{1}{4^{z+1}} − \frac{1}{12^{z+1}} − \frac{1}{20^{z+1}} + \frac{1}{60^{z+1}} ) \frac{1}{(1 − 3^{−z})(1 − 5^{−z})}.

Notice that each term of the form

\frac{b^{−z}}{(1 − 3^{−z})(1 − 5^{−z})}

with b > 1 contributes

S_{N} = ∑_{e_{1}=1}^{⌊log_{3}(N/b)⌋} \underbrace{∑_{e_{2}=0}^{⌊log_{5}(N/b3^{e_{1}})⌋} 1}_{= \frac{log N}{log 5} − \frac{log 3}{log 5} e_{1}+O(1)}

= \frac{log N}{log 5} ∑_{e_{1}=1}^{⌊log_{3}(N/b)⌋} 1 − \frac{log 3}{log 5} ∑_{e_{1}=1}^{⌊log_{3}(N/b)⌋} e_{1} + O(log N)

= \frac{(log N)^{2}}{log 3 log 5} − \frac{log 3}{2 log 5}⌊log_{3}(N/b)⌋^{2} + O(log N)
= \frac{(log N)^{2}}{log 3 log 5} (1 − \frac{1}{2}) + O(log N)

to π_{α}(N). Summing over all the terms therefore gives

π_{α}(N) = \frac{15}{4} (1 − \frac{1}{3} − \frac{1}{5} + \frac{1}{15}) \frac{(log N)^{2}}{2 log 3 log 5} + O(log N)
= \frac{(log N)^{2}}{log 3 log 5} + O(log N).

Returning to the general case, we will use an induction argument to obtain an exact asymptotic for S_{x}^{P} (0). The following lemma provides the essential inductive step, but is more general than we need at this stage (the full statement being needed for the proof of Theorem 3.6(3)). The result is a little technical and requires some preparation. Suppose m = (m_{p}) ∈ ℤ^{P} satisfies m_{p} ⩽ 0 for all p ∈ P. Set

\overline{P} = \{p ∈ P : m_{p} < 0\}.  (30)

Let \overline{P} ⊆ W ⊆ P, d ∈ ℕ_{0}^{W} and define d̃ = (d̃_{p}) ∈ ℕ_{0}^{W} by

d̃_{p} = \begin{cases}0 & if p ∈ \overline{P} , \\ d_{p} & if p ∉ \overline{P} . \end{cases}  (31)

Set

T_{x}^{W} (m, d, k) = (log x)^{−k} ∑_{e∈ℕ_{0}^{W} ,} ; φ_{W} (e)⩽xφ_{W} (m′e)e^{d},  (32)

where m′ = (m_{p})_{p∈W}.

Notice that

T_{x}^{W} (m, d, k) ⩽ (log x)^{−k} ∏_{p∈W} ∑_{e_{p}=0}^{⌊log_{p} x⌋} p^{m_{p}e_{p}}e_{p}^{d_{p}}
⩽ C(log x)^{−k} ∏_{p∈W \\overline{P}}(log x)^{d_{p}+1}

= C(log x)^{d̃·1+|W \\overline{P} |−k}  (33)

Lemma 7.3. Suppose that W ≠ \overline{P} and W′ = W \ \{s\} for some s ∈ W \ \overline{P}. If m̃ = 0, d = d̃, k = d · 1 + |W \ \overline{P} | then there exist a finite indexing set H and set of constants {c_{h} : h ∈ H}, both independent of x, such that

T_{x}^{W} (m, d, k) = (∑_{h∈H} c_{h}T_{x}^{W′}(m, d(h), k_{h})) + O (\frac{1}{log x}) ,

where for each h ∈ H, d(h) ∈ ℕ_{0}^{W′} and k_{h} ∈ ℕ_{0} satisfy

(1) d(h) = d̃(h),

(2) k_{h} = d(h) · 1 + |W′ \ \overline{P} |.

Proof. If W′ = ∅ the result is obvious, so assume W′ ≠ ∅. By assumption m_{s} = 0, so

T_{x}^{W} (m, d, k) = (log x)^{−k} ∑_{e∈ℕ_{0}^{W′},} ; φ_{W′}(e)⩽xφ_{W′}(m′e)e^{d′} ∑_{e_{s}=0}^{⌊log_{s}(x/φ_{W′}(e))⌋} e_{s}^{d_{s}}, (34)

where m′ = (m_{p})_{p∈W′} and d′ = (d_{p})_{p∈W′}. Expanding the inner sum,

∑_{e_{s}=0}^{⌊log_{s}(x/φ_{W′}(e))⌋} e_{s}^{d_{s}} = C⌊log_{s} x/φ_{W′}(e)⌋^{d_{s}+1} + O ((log x)^{d_{s}})
= C (log_{s} x/φ_{W′}(e))^{d_{s}+1} + O ((log x)^{d_{s}})
= C (log x − e · (log p)_{p∈W′})^{d_{s}+1} + O ((log x)^{d_{s}})
= ∑_{h∈H} c_{h}(log x)^{k′_{h}}e^{d′(h)} + O ((log x)^{d_{s}}) ,  (35)

where the finite indexing set H and set of constants {c_{h} : h ∈ H} are both independent of x, and k_{h} ∈ ℕ_{0}, d′(h) ∈ ℕ_{0}^{W′} satisfy

k′_{h} + d′(h) · 1 = d_{s} + 1  (36)

Let k_{h} = k − k′_{h} and d(h) = d′ + d′(h). Substituting (35) into (34) gives

T_{x}^{W} (m, d, k) = ( ∑_{h∈H′} c_{h}T_{x}^{W′}(m, d(h), k_{h})) + O (\frac{1}{log x}); (37)

the error term being obtained using (33), since

(log x)^{d_{s}−k} ∑_{e∈ℕ_{0}^{W′},} ; φ_{W′}(e)⩽xφ_{W′}(m′e)e^{d′} = O ((log x)^{d̃′·1+|W′\\overline{P} |+d_{s}−k})

= O ((log x)^{d̃·1+|W \\overline{P} |−k−1})
= O ((log x)^{k−k−1}) .

It remains to see that conditions (1) and (2) are satisfied. To see that (2) holds, note (36) implies

k − k_{h} + (d(h) − d′) · 1 = d_{s} + 1
⇒ k_{h} = d(h) · 1 + k − d · 1 − 1
= d(h) · 1 + |W′ \ \overline{P} |,

since by hypothesis k − d · 1 = |W \ \overline{P} |.

To satisfy condition (1), we show that H may simply be replaced by

{h ∈ H : d(h) = d̃(h)},

in (37). To see this, note that if d(h) ≠ d̃(h) then d̃(h)·1 ⩽ d(h)·1−1 and so

d̃(h) · 1 + |W′ \ \overline{P} | − k_{h} ⩽ d(h) · 1 + |W′ \ \overline{P} | − k_{h} − 1
⩽ −1,

since (2) holds. Therefore,

T_{x}^{W′}(m, d(h), k_{h}) = O (\frac{1}{log x})

by (33). □

Let L = |P |. Since

(log x)^{−L}S_{x}^{P} (0) = T_{x}^{W′}(0, 0, L),

applying Lemma 7.3 and induction, we obtain

(log x)^{−L}S_{x}^{P} (0) = C_{1} + O (\frac{1}{log x})

for some constant C_{1} which is independent of x. Notice that C_{1} > 0, otherwise there would be a contradiction to the lower bound provided by Lemma 7.1. Furthermore,

S_{N/b_{I}}^{P}(0) = C_{1}(log(N/b_{I}))^{L} + O((log(N/b_{I}))^{L−1})
= C_{1}(log N)^{L} + O((log N)^{L−1})

It follows that the contribution to π_{α}(N) from (28) is

C_{2} (∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}} ) (log N)^{L} + O((log N)^{L−1}),

where C_{2} > 0 and also ∑_{I⊂J}(−1)^{|I|}/c_{I} > 0.

Since there is at least one term of the form (28) contributing to π_{α}(N) for which L = K, this completes the proof.

7.2. Proof of Theorem 3.6(2). Recall that in this case, for each term of the form (28) contributing to π_{α}(N), n = (n_{p}) satisfies

max{n_{p} : p ∈ P } ⩽ σ.

Moreover, since σ is a simple pole, any term with max{n_{p} : p ∈ P } = σ has at most one of the corresponding n_{p} equal to σ. The upper bound now follows from Lemma 7.1.

For the lower bound we use more elementary methods. The orbits for these systems mainly accumulate at specific lengths, and summing over orbits of those lengths suffices. Recall that O_{α}(n) is related to the number of periodic points by (4) where F_{α}(n) is given by the formula (8) with S = T. By (23), there is a prime q ∈ Q such that

∑_{v∈U} r_{v} = σ + 1,  (38)

where U = {v ∈ T: v|q}.

Using the notation from Section 5, let ℓ = lcm{ℓ_{v}: v ∈ U}. Notice that gcd(q, ℓ) = 1 since ℓ_{v} divides q^{r_{v}} − 1 for v ∈ U. Using basic properties of the the Möbius function,

O_{α}(ℓq^{e}) = \frac{1}{ℓq^{e}} ∑_{e′=0}^{e} ∑_{d|ℓ} μ(q^{e−e′}ℓ/d)f_{T} (dq^{e′})

= \frac{1}{ℓq^{e}} ∑_{d|ℓ} μ(ℓ/d) (f_{T} (dq^{e}) − f_{T} (dq^{e−1}))

= \frac{1}{ℓq^{e}} (f_{T} (ℓq^{e}) − f_{T} (ℓq^{e−1}))

+\frac{1}{ℓq^{e}} ∑_{d|ℓ,d<ℓ} μ(ℓ/d) (f_{T} (dq^{e}) − f_{T} (dq^{e−1})) .

By (38) and Lemma 2.6, there are constants D, E > 0 such that

f_{T} (ℓq^{e}) = Dq^{(σ+1)e}  and  f_{T} (dq^{e}) ⩽ Eq^{σe}

for all d < ℓ and large enough e. Now,

O_{α}(ℓq^{e}) ⩾ \frac{1}{ℓq^{e}} ( (Dq^{(σ+1)e} − Dq^{(σ+1)(e−1)}) − 2Eℓq^{σe})
= \frac{D}{ℓ} (1 − q^{−(σ+1)}) q^{σe} − 2Eq^{(σ−1)e},

so there are constants C_{1} ⩾ 0 and C_{2}, C_{3}, C_{4} > 0 such that

π_{α}(N) ⩾ ( ∑_{e=0}^{⌊log_{q} N/ℓ⌋} O_{α}(ℓq^{e})) − C_{1}

⩾ ( ∑_{e=0}^{⌊log_{q} N/ℓ⌋} C_{2}q^{σe} − C_{3}q^{(σ−1)e}) − C_{1}
⩾ C_{4}N^{σ},

for all large N.

7.3. Proof of Theorem 3.6(3). We begin with an example that illuminates some of the issues that arise in this case.

Example 7.4. Consider M = ℤ_{(3)}^{2} × ℤ_{(5)}^{2}, X = M̂ and the the endomorphism α : X → X given by x ↦ 2x. The dynamical system (X, α) has periodic point data identical to that of a product of Example 4.2 and another duplicate system. Consequently,

F_{α}(n) = \begin{cases}1 & if n is odd; \\ 9 · 3^{2e_{1}} & if n = 2 · k · 3^{e_{1}}, 2 ∤ k, 3 ∤ k; \\ 225 · 3^{2e_{1}} · 5^{2e_{2}} & if n = 4 · k · 3^{e_{1}} · 5^{e_{2}}, 3 ∤ k, 5 ∤ k. \end{cases}

Following a similar method to that used in Example 4.2, the term of d_{α} dominating the growth of π_{α} is

(\frac{1}{4^{z+1}} − \frac{1}{12^{z+1}} − \frac{1}{20^{z+1}} + \frac{1}{60^{z+1}} ) \frac{225}{(1 − 3^{1−z})(1 − 5^{1−z})}.  (39)

Hence, consider

S_{N/b} = ∑_{e∈ℕ_{0}^{2},} ; 3^{e_{1}}5^{e_{2}}⩽N/b3^{e_{1}}5^{e_{2}}.

Then, writing {·} for the fractional part,

\frac{S_{N/b}}{N log N} = \frac{1}{N log N} ∑_{e∈ℕ_{0}^{2},} ; 3^{e_{1}}5^{e_{2}}⩽N/b3^{e_{1}}5^{e_{2}}

= \frac{1}{N log N} ∑_{e_{1}=0}^{⌊log_{3}(N/b)⌋} 3^{e_{1}} ∑_{e_{2}=0}^{⌊log_{5}(N/3^{e_{1}}b)⌋} 5^{e_{1}}

= \frac{1}{N log N} ∑_{e_{1}=0}^{⌊log_{3} N/b⌋} 3^{e_{1}} (\frac{5N}{4b} 3^{−e_{1}}5^{−\{log_{5}(N/3^{e_{1}}b)\}} − \frac{1}{4})

= \frac{5}{4b log N} ∑_{e_{1}=0}^{⌊log_{3} N/b⌋} 5^{−\{log_{5} N/b−e_{1} log_{5} 3\}} + O(N^{−1}).

Since the exponent in the inner sum is uniformly distributed in the unit interval, by Weyl’s theorem [18],

\frac{1}{log N} ∑_{e_{1}=0}^{⌊log_{3} N/b⌋} 5^{−\{log_{5} N/b−e_{1} log_{5} 3\}} → \frac{1}{log 3} ∫_{0}^{1} 5^{−y} dy

= \frac{4}{5 log 3 log 5}.

Therefore,

\frac{S_{N/b}}{N log N} → \frac{1}{b log 3 log 5}.

From the above and (39), it follows that

\frac{π_{α}(N)}{N log N} ∼ \frac{225}{N log N} (\frac{S_{N/4}}{4} − \frac{S_{N/12}}{12} − \frac{S_{N/20}}{20} + \frac{S_{N/60}}{60} )
∼ \frac{225}{log 3 log 5} (\frac{1}{4^{2}} − \frac{1}{12^{2}} − \frac{1}{20^{2}} + \frac{1}{60^{2}} )
= \frac{12}{log 3 log 5}.

In the general case, the proof uses similar ideas, with the main steps being equidistribution and an inclusion-exclusion argument. As in the previous section, for each term of the form (28) contributing to π_{α}(N), n = (n_{p}) satisfies

max{n_{p} : p ∈ P } ⩽ σ,

and if max{n_{p} : p ∈ P } = σ then

|{p ∈ P : n_{p} = σ}| ⩽ K.

Furthermore, there is at least one term with

max{n_{p} : p ∈ P } = σ and |{p ∈ P : n_{p} = σ}| = K  (40)

In light of the upper bound provided by Lemma 7.1, terms of this form give the dominant contribution to π_{α}(N). Thus, to obtain the required asymptotic it suffices to prove that

\frac{1}{N^{σ}(log N)^{K−1}} ∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}} S_{N/b_{I}}^{P}(n)  (41)

convergences to a positive constant when n satisfies (40).

Choose a prime q ∈ P with n_{q} = σ and set P′ = P \ {q}. Write

a_{N} = \frac{1}{N^{σ}(log N)^{K−1}} S_{N/b}^{P}(n).

Then

a_{N} = \frac{1}{N^{σ}(log N)^{K−1}} ∑_{φ_{P′}(e)⩽N/b}φ_{P′}(n′e) ∑_{e_{q}=0}^{⌊log_{q}(N/bφ_{P′}(e))⌋}q^{σe_{q}}

where n′ = (n_{p})_{p∈P′}. Treating the inner sum as a geometric progression as usual, we see that a_{N} is equal to

\frac{C_{1}}{b^{σ}(log N)^{K−1}} ( ∑_{φ_{P′}(e)⩽N/b}φ_{P′}(n′e)φ_{P′}(σe)^{−1}q^{−σ\{log_{q}(N/bφ_{P′}(e))\}}) + Δ_{N}

where C_{1} = 1/(1 − q^{−σ}) and

Δ_{N} = O (\frac{1}{N^{σ}(log N)^{K−1}} S_{N/b}^{P′}(n′)) = O (\frac{1}{log N} )

by Lemma 7.1.

Now let r ∈ P′ be a prime with m_{r} = σ (such a prime exists by hypothesis) and set W = P \ {q, r}. Then a_{N} is, up to O(1/ log N),

\frac{C_{1}}{b^{σ}(log N)^{K−1}} ∑_{φ_{W} (e)⩽N/b}φ_{W} (me) ∑_{e_{r}=0}^{⌊log_{r}(N/bφ_{W} (e))⌋} q^{−σ\{log_{q}(N/bφ_{W} (e)r^{er})\}}

where m = (m_{p}) ∈ ℤ^{W} is given by

m_{p} = \begin{cases}n_{p} − σ & if n_{p} < σ, \\ 0 & if n_{p} = σ. \end{cases}

For φ_{W} (e) ⩽ x, set

I_{x}(e) = \frac{1}{log_{r}(x/φ_{W} (e))} ∑_{e_{r}=0}^{⌊log_{r}(x/φ_{W} (e))⌋} q^{−σ\{log_{q}(x/φ_{W} (e)r^{er})\}}

and

J_{x}(e) = φ_{W} (me) log(x/φ_{W} (e)).

Then, again up to O(1/ log N), a_{N} is

\frac{C_{2}}{b^{σ}(log N)^{K−1}} ∑_{φ_{W} (e)⩽N/b}J_{N/b}(e)I_{N/b}(e),

where C_{2} = C_{1}/ log r. Since r ≠ q the circle rotation by log_{q} r is uniquely ergodic, so there is uniform convergence for continuous functions in the ergodic theorem (see Oxtoby [10, §5] for example). Hence, given ε > 0 there is an M such that whenever N/bφ_{W} (e) ⩾ M,

I_{N/b}(e) = ∫_{0}^{1} q^{−σy} dy + δ_{N/b}(e),

where |δ_{N/b}(e)| < ε. For N/bφ_{W} (e) < M (that is in the range N/bM < φ_{W} (e) ⩽ N/b) there exists a uniform constant D for which

I_{N/b}(e) = ∫_{0}^{1} q^{−σy} dy + β_{N/b}(e)

where |β_{N/b}(e)| < D.

Let

b_{N} = ∑_{φ_{W} (e)⩽N/b}J_{N/b}(e),
c_{N} = ∑_{φ_{W} (e)⩽N/bM} J_{N/b}(e)δ_{N/b}(e), and
d_{N} = ∑_{N/bM<φ_{W} (e)⩽N/b}J_{N/b}(e)β_{N/b}(e).

Then

a_{N} = \frac{1}{b^{σ}} (log N)^{1−K}(C_{3}b_{N} + C_{2}c_{N} + C_{2}d_{N}) + O (\frac{1}{log N} ) ,

where C_{3} > 0 is independent of b.

Lemma 7.5. We have the following estimates:

(1) (log N)^{1−K}b_{N} converges to a constant C_{4} ⩾ 0 that is independent of b;

(2) (log N)^{1−K}|c_{N}| ⩽ C_{5}ε; and

(3) (log N)^{1−K}d_{N} converges to zero as N → ∞.

Assuming Lemma 7.5 for now and setting C_{6} = C_{3}C_{4}, we deduce that

a_{N} ⟶ \frac{C_{6}}{b^{σ}} as N → ∞  (42)

This can be used to show that (41) converges to a positive constant, therefore completing the proof. To see this, first note that if C_{6} = 0 then (42) contradicts the lower bound provided by Lemma 7.1, therefore C_{6} > 0. Furthermore, (41) converges to

C_{6} ∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}b_{I}^{σ}} = C_{6} ∑_{I⊂J} \frac{(−1)^{|I|}}{c_{I}(Bc_{I})^{σ}}

= \frac{C_{6}}{B^{σ}} ∑_{I⊂J′} \frac{(−1)^{|I|}}{lcm\{i: i ∈ I\}}
> 0,

where J′ = {i^{σ+1} : i ∈ J}.

Finally we turn to the postponed proof of the lemma.

Proof of Lemma 7.5. To prove (1), first note we may write

(log N)^{1−K}b_{N} = \frac{1}{(log(N/b) + log b)^{K−1}} b_{N}
= (1 + Δ_{N})^{−1}(log(N/b))^{1−K}b_{N}

where Δ_{N} = O((log N)^{−1}). Therefore, it suffices to prove

(log(N/b))^{1−K}b_{N} → C_{4}.

Write x = N/b and consider

(log(N/b))^{1−K}b_{N} = (log x)^{1−K} ∑_{φ_{W} (e)⩽x}J_{x}(e)
= (log x)^{1−K} ∑_{φ_{W} (e)⩽x}φ_{W} (me) log(x/φ_{W} (e))
= (log x)^{1−K} ∑_{φ_{W} (e)⩽x}φ_{W} (me)(log x − e · (log p)_{p∈W}).

Therefore, (log(N/b))^{1−K}b_{N} may be written as a sum of |W | + 1 expressions of the form

(log x)^{−k} ∑_{φ_{W} (e)⩽x}φ_{W} (me)e^{d},

where d ∈ ℕ_{0}^{W} satisfies

k − d · 1 = K − 2 = |W \ \overline{P} |,  (43)

with \overline{P} defined by (30).

So, we must consider an expression of the form T_{x}^{W} (m, d, k), defined by (32). Let d̃ be defined by (31). If d ≠ d̃ then d̃ · 1 < d · 1. Hence by (33) and (43),

T_{x}^{W} (m, d, k) = O (\frac{1}{log x}) .

Thus, we need only consider those terms T_{x}^{W} (m, d, k) with d = d̃. If \overline{P} = ∅, then proceeding as in Section 7.1, an induction using Lemma 7.3 shows that

T_{x}^{W} (m, d, k) = C_{7} + O (\frac{1}{log x}) ,

where C_{7} is a constant independent of x. If \overline{P} ≠ ∅ then again by induction using Lemma 7.3, up to an error of O(1/ log x), T_{x}^{W} (m, d, k) may be written as a linear combination of expressions of the form

T \overline{P} ; x (m, 0, 0) = ∑_{φ\overline{P} (e)⩽x}φ\overline{P} (m′e)

where m′ = (m_{p})_{p∈\overline{P}}. Moreover,

∏_{p∈\overline{P}} ∑_{p=0}^{⌊|\overline{P} |^{−1} log_{p} x⌋} p^{m_{p}e_{p}} ⩽ T \overline{P} ; x (m, 0, 0) ⩽ ∏_{p∈\overline{P}} ∑_{p=0}^{⌊log_{p} x⌋} p^{m_{p}e_{p}},

so

T \overline{P} ; x (m, 0, 0) ⟶ ∏_{p∈\overline{P}} \frac{1}{1 − p^{m_{p}}} .

as x → ∞.

Thus, (log(N/b))^{1−K}b_{N} converges to a constant C_{4} independent of b which is non-negative, as each term of the sequence is non-negative.

To prove (2), simply note that |c_{N}| ⩽ b_{N}ε. To prove (3), first notice

|d_{N}| ⩽ D(log N)^{1−K} ∑_{N/bM<φ_{W} (e)⩽N/b}J_{N/b}(e)

= D(log N)^{1−K} (b_{N} − ∑_{φ_{W} (e)⩽N/bM} J_{N/b}(e)) .

But

J_{N/b}(e) = J_{N/bM}(e) + (log M)φ_{W} (me),

so

|d_{N}| ⩽ D(log N)^{1−K}(b_{N} − b_{N/M}) − D(log M)(log N)^{1−K}S_{N/bM}^{W}(m)
= \frac{D}{(log N)^{1−K}} (b_{N} − b_{N/M}) + O (\frac{1}{log N} ) ,

by Lemma 7.1. Since (log N)^{1−K}b_{N} and (log N)^{1−K}b_{N/M} both converge to C_{4}, (3) follows. □

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School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

E-mail address: t.ward@uea.ac.uk