On density and multiplicative structure of sets of generalized integers

Štefan Porubský Institute of Computer Science Academy of Sciences of the Czech Republic Pod Vodárenskou věží 2 182 07 Prague 8

      email: Stefan.Porubsky@cs.cas.cz

Porubský, Š.: Notes on density and multiplicative structure of sets of generalized integers, in: Topics in Classical Number Theory, Colloquia Mathematica Societatis János Bolyai, 34., Budapest 1984, pp. 1295 – 1315

A ⊂ ℕ,  N_{A}(x) = ∑_{\begin{matrix}a_{i}∈A \\ a_{i}≤x \end{matrix}} 1,  x > 1

• asymptotic density:

\underline{d}(A) = lim inf_{x→∞} \frac{N_{A}(x)}{x} , \overline{d}(A) = lim sup_{x→∞} \frac{N_{A}(x)}{x}

• logarithmic density:

\underline{ℓ}(A) = lim inf_{x→∞} \frac{1}{log x} ∑_{\begin{matrix}a_{i}∈A \\ a_{i}≤x \end{matrix}} \frac{1}{a_{i}}, \overline{ℓ}(A) = lim sup_{x→∞} \frac{1}{log x} ∑_{\begin{matrix}a_{i}∈A \\ a_{i}≤x \end{matrix}} \frac{1}{a_{i}}

• Schnirelmann density: h(A) = inf_{x} \{\frac{N_{A}(x)}{x}\}

0 ≤ \underline{d}(A) ≤ \underline{ℓ}(A) ≤ \overline{ℓ}(A) ≤ \overline{d}(A)

\underline{Common features}: a. density is a non–negative real number b. finite sets have zero density c. if A ⊂ B ⊂ ℕ then density of A does not exceed

density of B Arithmetical semigroups

• (𝔾, .) free commutative semigroup with identity element 1_{𝔾}

• P_{𝔾} ≤ ∞ the set of generators (the so–called primes)

• norm | · | : 𝔾 → ℝ:

⋆ |1_{𝔾}| = 1, |a| > 1 for all a ∈ 𝔾,

⋆ |ab| = |a|.|b| for all a, n ∈ 𝔾,

⋆ N_{𝔾}(x) = ∑\begin{matrix}|a|≤x \\ a∈𝔾 \end{matrix} 1 < ∞ for each real x.

Axiom A: There exists positive constants A and δ and a constant η with 0 ≤ η < δ, such that

N_{𝔾}(x) = Ax^{δ} + O(x^{η})

ζ–function of 𝔾: ζ_{𝔾}(s) = ∑_{a∈𝔾} \frac{1}{|a|^{s}}

Lemma. Let 𝔾 be an arithmetical semigroup satisfying Axiom A. Then ∑_{a∈𝔾} |a|^{−z} is absolutely convergent for ℜ(z) > δ, and divergent for ℜ(z) ≤ δ. Moreover

∑_{|a|≤x} |a|^{−δ} = δA log x + γ_{𝔾} + O(x^{η−δ}).

Example 1. 𝔾 = ℕ, the set of positive integers where

N_{ℕ}(x) = x + O(1).

Example 2. 𝔾 = G_{K}, the semigroup of all non–zero integral ideals in a given algebraic number field K of degree n = [K : ℚ] over rationals ℚ with the usual norm function |𝔞| = card(O_{K}/𝔞). Then

N_{K}(x) = A_{K}x + O (x^{\frac{n−1}{n+1}}) ,

where A_{K} can be explicitly given. Example 3. 𝔾 = 𝔸 the category of all finite Abelian groups with the usual direct product operation and the norm |H| = card(H). Fundamental Theorem on finite Abelian groups shows that 𝔸 is free and that the generators are the cyclic groups of prime–power order. The fact that this arithmetical semigroup satisfies Axiom A follows from an older result of Erdős and Szekeres that

N_{A}(x) = αx + O (\sqrt{x}) ,

where α = ∏_{j=1}^{∞} ζ_{ℕ}(js) with ζ_{ℕ} = ζ denoting the classi-

cal Riemann zeta function. δ ∈ (−∞, +∞)

δ–regularly varying function function F (x) defined and measurable on ⟨0, ∞):

lim_{x→∞} \frac{F (λx)}{F (x)} = λ^{δ} if for every λ > 0

If F (x) = x^{δ}L(x), then L(x) is slowly oscillating (i.e. δ = 0).

Arithmetical semigroup 𝔾 will be called δ–regular if the counting function N_{𝔾}(x) is δ–regularly varying function.

Lemma. Let 𝔾 be a δ–regular semigroup. Then ∑_{a∈𝔾} |a|^{−z} is convergent for all ℜ(z) > δ and divergent for all ℜ(z) < δ.

Wegmann (1966): If N_{𝔾}(x) ∼ \frac{x}{log^{2} x} (i.e. 𝔾 is 1–

regular), then ∑_{a∈𝔾} |a|^{−1} < ∞. 𝔪 : 𝔾 → ℝ, C ⊂ 𝔾: χ_{C} indicator of C N_{C}(𝔪, x) = ∑_{|a|≤x} 𝔪(a)χ_{C}(a) N′_{C}(𝔪, x) = ∑_{|a|=x} 𝔪(a)χ_{C}(a)

σ_{x}(C, 𝔪) = \frac{∑_{|a|≤x} 𝔪(a)χ_{C}(a)}{∑_{|a|≤x} 𝔪(a)} = \frac{N_{C}(𝔪, x)}{N_{𝔾}(𝔪, x)}

lower 𝔪–density: \underline{σ}(C, 𝔪) = lim inf_{x→∞} σ_{x}(C, 𝔪) upper 𝔪–density: \overline{σ}(C, 𝔪) = lim sup_{x→∞} σ_{x}(C, 𝔪)

A. 𝔪 is non–negative, i.e. 𝔪(a) ≥ 0 for every a ∈ 𝔾 B. ∑_{a∈𝔾} 𝔪(a) diverges

A implies a B implies b B implies the summation method is regular Knopp’s theorem on convergence kernel: Theorem. Let 𝔪 and 𝔰 be two positive functions defined on an arithmetical semigroup 𝔾 such that

(i) the series ∑_{a∈𝔾} 𝔰(a) diverges

(ii) lim_{x→∞} \frac{N′_{𝔾}(𝔪, x)}{N_{𝔾}(𝔪, x)} = lim_{x→∞} \frac{∑_{|a|=x} 𝔪(a)}{∑_{|a|≤x} 𝔪(a)} = 0

lim_{x→∞} \frac{N′_{𝔾}(𝔰, x)}{N_{𝔾}(𝔰, x)} = lim_{x→∞} \frac{∑_{|a|=x} 𝔰(a)}{∑_{|a|≤x} 𝔰(a)} = 0

(iii) if a_{1}, a_{2} ∈ 𝔾 be such that |a_{1}| ≤ |a_{2}| then

\frac{𝔪(a_{2})}{𝔪(a_{1})} ≥ \frac{𝔰(a_{2})}{𝔰(a_{1})}.

Then \underline{σ}(C, 𝔪) ≤ \underline{σ}(C, 𝔰) ≤ \overline{σ}(C, 𝔰) ≤ \overline{σ}(C, 𝔪)

for every C ⊂ 𝔾.

If N_{𝔾}(𝔪, x) is δ–regular, (ii) is superfluous:

∑_{|a|≤x} 𝔪(n) = x^{δ}L(x), and 0 < α < 1 arbitrary

0 ≤ \frac{∑_{|a|=x} 𝔪(n)}{∑_{|a|≤x} 𝔪(n)} ≤ \frac{∑_{αx<|a|≤x} 𝔪(n)}{∑_{|a|≤x} 𝔪(n)}

= \frac{x^{δ}L(x) − α^{δ}x^{δ}L(αx)}{x^{δ}L(x)} → 1 − α^{δ}

M. to every a ∈ 𝔾 there exists a positive real number 𝔪̂(a), 𝔪̂ < 1 such that for every subset C ⊂ 𝔾 having the 𝔪–density σ(C, 𝔪) the set aC = {ac : c ∈ 𝔾} has also the 𝔪–density and

σ(aC, 𝔪) = 𝔪̂(a)σ(C, 𝔪).

• if 𝔪(a) = 1 for every a ∈ 𝔾 then 𝔪̂(a) = \frac{1}{a}

• if 𝔪 is completely multiplicative and N_{𝔾}(𝔪, x) = x^{Δ}L(x), where L(x) is slowly oscillating, then

𝔪̂(a) = 𝔪(a)|a|^{−Δ}.

\underline{Question:}

Under which conditions does 𝔪̂ fulfil the conditions A and B?

Theorem. Let 𝔾 be an arithmetical semigroup. Let 𝔪 satisfy conditions A, B, and M. Let in the case, when

∑_{p∈P_{𝔾}} 𝔪̂(p) < ∞, (∗)

we have uniformly in x and p ∈ P_{𝔾}

σ_{p𝔾}(𝔪, x) = O(𝔪̂(p)) (∗∗)

Then 𝔪̂ fulfils conditions A and B.

Note: (∗∗) cannot be omitted if (∗) holds!

Take Wegmann’s 𝔾 with asymptotic density. Then (∗) holds (𝔪̂(p) = |p|^{−1}), while it can be shown that the finite set 𝔾⟨P_{𝔾}⟩ = {a ∈ 𝔾 : p ∤ a for every p ∈ P_{𝔾}} = {1_{𝔾}} has non–zero density

∏_{p∈P_{𝔾}}(1 − |p|^{−1}).

Lemma. If the arithmetical semigroup 𝔾 satisfies Axiom A then the series

∑_{p∈P_{𝔾}} |p|^{−δ}

diverges.

We have

• 𝔪̂ is completely multiplicative (ab𝔾 = a(b𝔾)),

• lim_{|a|→∞} 𝔪̂(a) = 0 (if lim_{|p|→∞} 𝔪̂(p) = 0).

Therefore

• a ↦ \frac{1}{𝔪̂(a)} is a norm on 𝔾

• ζ̂_{𝔾}(z) = ∑_{a∈𝔾} (\frac{1}{𝔪̂(a)})^{−z} = ∑_{a∈𝔾} 𝔪̂(a)^{z}

B says that ζ̂_{𝔾}(z) has a pole at z = 1

if 𝔪(a) = 1 for every a ∈ 𝔾 and 𝔾 is δ–regular, then ζ̂_{𝔾}(z) = ζ_{𝔾}(sδ), and therefore B holds if 𝔾 satisfies Axiom A. slowly oscillating function L(x) is called good if lim_{x→∞} Z(x) = ∞, where Z(x) = ∫_{1}^{x} L(y)y^{−1}dy.

if lim inf_{x→∞} L(x) > 0, then L is good

Theorem. Let 𝔪 be a completely multiplicative function defined on an arithmetical semigroup 𝔾. Let

N_{𝔾}(𝔪, x) = ∑_{|a|≤x} 𝔪(a) = x^{δ}L(x),

with L(x) a good slowly oscillating function. Then

0 ≤ \underline{σ}(C, 𝔪) ≤ \underline{σ}(C, 𝔪̂) ≤ \overline{σ}(C, 𝔪̂) ≤ \overline{σ}(C, 𝔪) ≤ 1

for every C ⊂ 𝔾.

Corollary. Let 𝔪 be a positive completely multiplicative function defined on the arithmetical semigroup 𝔾 such that

∑_{|a|≤x} 𝔪(a) = L(x),

where L is a good slowly oscillating function, then

\underline{σ}(C, 𝔪) = \underline{σ}(C, 𝔪̂), and \overline{σ}(C, 𝔪̂) = \overline{σ}(C, 𝔪)

for every C ⊂ 𝔾.

Corollary. Let 𝔪 be a completely multiplicative function defined on an arithmetical semigroup 𝔾. Let

N_{𝔾}(𝔪, x) = ∑_{|a|≤x} 𝔪(a) = x^{δ}L(x),

with L(x) a good slowly oscillating function. Then

ˆ𝔪̂ = 𝔪̂.

Corollary. Let the arithmetical semigroup 𝔾 satisfies Axiom A. Let 𝔪 be a positive function defined on 𝔾 such that

• the series ∑_{a∈𝔾} 𝔪(a) diverges,

• lim_{x→∞} \frac{N′_{𝔾}(𝔪, x)}{N_{𝔾}(𝔪, x)} = lim_{x→∞} \frac{∑_{|a|=x} 𝔪(a)}{∑_{|a|≤x} 𝔪(a)} = 0,

• |a_{1}| ≤ |a_{2}| ⇒ 𝔪(a_{1}) ≤ 𝔪(a_{2})

then the lower and upper 𝔪–density coincides with the lower and upper logarithmic density.

Theorem. Let 𝔪 : 𝔾 → ℝ^{+} be a completely multiplicative function such that

N_{𝔾}(𝔪, x) = Bx^{Δ} + O(x^{Θ}), 0 ≤ Θ < Δ as x → ∞, then

N_{𝔾}(𝔪̂, x) = ΔB log x + ψ_{m} + O(x^{Θ−Δ}) with a suitable ψ_{m}. lower 𝔪–density:

\underline{σ}_{𝔾}(C, 𝔪) = lim inf_{x→∞} \frac{N_{C}(𝔪, x)}{N_{𝔾}(𝔪, x)} = lim inf_{x→∞} \frac{∑_{a∈C,|a|≤x} 𝔪(a)}{Bx^{Δ}}

upper 𝔪–density:

\overline{σ}_{𝔾}(C, 𝔪) = lim sup_{x→∞} \frac{N_{C}(𝔪, x)}{N_{𝔾}(𝔪, x)} = lim sup_{x→∞} \frac{∑_{a∈C,|a|≤x} 𝔪(a)}{Bx^{Δ}}

lower 𝔪̂–density:

\underline{d}_{𝔾}(C) = lim inf_{x→∞} \frac{N_{C}(𝔪̂, x)}{N_{𝔾}(𝔪̂, x)} = lim inf_{x→∞} \frac{∑_{a∈C,|a|≤x} 𝔪(a)|a|^{−Δ}}{ΔB log x}

upper 𝔪̂–density:

\overline{d}_{𝔾}(C) = lim sup_{x→∞} \frac{N_{C}(𝔪̂, x)}{N_{𝔾}(𝔪̂, x)} = lim sup_{x→∞} \frac{∑_{a∈C,|a|≤x} 𝔪(a)|a|^{−Δ}}{ΔB log x}

if Axiom A holds for 𝔾 then ∑_{|a|≤x} |a|^{−δ} ∼ δA log x, and

N_{𝔾}(x) ∼ Ax^{δ}

lower logarithmic density:

\underline{ℓ}_{𝔾}(C) = lim inf_{x→∞} \frac{1}{δA log x} ∑_{|a|≤x} |a|^{−δ}

upper logarithmic density:

\overline{ℓ}_{𝔾}(C) = lim sup_{x→∞} \frac{1}{δA log x} ∑_{|a|≤x} |a|^{−δ}

lower asymptotic density:

\underline{d}_{𝔾}(C) = lim inf_{x→∞} \frac{N_{C}(1, x)}{N_{𝔾}(x)} = lim inf_{x→∞} \frac{∑_{a∈C,|a|≤x} 1}{Ax^{δ}}

upper asymptotic density:

\overline{d}_{𝔾}(C) = lim sup_{x→∞} \frac{N_{C}(1, x)}{N_{𝔾}(x)} = lim sup_{x→∞} \frac{∑_{a∈C,|a|≤x} 1}{Ax^{δ}}

Corollary. Let 𝔪 : 𝔾 → ℝ^{+} be a completely multiplicative function such that

N_{𝔾}(𝔪, x) = Bx^{Δ} + O(x^{Θ}), 0 ≤ Θ < Δ

as x → ∞, then

0 ≤ \underline{σ}_{𝔾}(C, 𝔪) ≤ \underline{σ}_{𝔾}(C, 𝔪̂) ≤ \overline{σ}_{𝔾}(C, 𝔪̂) ≤ \overline{σ}_{𝔾}(C, 𝔪) ≤ 1

for every C ⊂ 𝔾.

Corollary. If the arithmetical semigroup 𝔾 satisfies Axiom A, then

0 ≤ \underline{d}_{𝔾}(C) ≤ \underline{ℓ}_{𝔾}(C) ≤ \overline{ℓ}_{𝔾}(C) ≤ \overline{d}_{𝔾}(C) ≤ 1

for every C ⊂ 𝔾.

Thank you!