ITERATION OF COMPOSITION OPERATORS ON SMALL BERGMAN SPACES OF DIRICHLET SERIES
arXiv:1705.05743v1 [math.CV] 16 May 2017
JING ZHAO
P −s A BSTRACT. The Hilbert spaces Hw consisiting of Dirichlet series F (s) = ∞ that satisn=1 a n n P∞ 2 fty n=1 |an | /w n < ∞, with {w n }n of average order log j n (the j -fold logarithm of n), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon– Hedenmalm theorem on such Hw from an iterative point of view. By that theorem, the composition operators are generated by functions of the form Φ(s) = c 0 s + φ(s), where c 0 is a nonnegative integer and φ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when c 0 = 0. It is verified for every integer j > 1, real α > 0 and {w n }n having average order (log+j n)α , that the composition operators map Hw into a scale of Hw ′ with
w n′ having average order (log+j +1 n)α . The case j = 1 can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.
1. INTRODUCTION Let H 2 be the Hilbert space of Dirichlet series with square summable coefficients. A theorem of Gordon and Hedenmalm [5] classifies the set of analytic functions Φ : C1/2 → C1/2 which generate composition operators that map H 2 into itself. Let Cθ denote the half-plane Cθ := {s = σ + i t : σ > θ}. The Gordon–Hedenmalm theorem reads as follows, in a slightly strengthened version found in [10]. Theorem 1 (Gordon–Hedenmalm Theorem). A function Φ : C1/2 → C1/2 generates a bounded composition operator CΦ : H 2 → H 2 if and only if Φ is of the form ∞ X cn n −s = c0 s + φ(s), (1) Φ(s) = c0 s + n=1
where c0 is a nonnegative integer, and φ is a Dirichlet series that converges uniformly in Cǫ (ǫ > 0) with the following mapping properties: (i) If c0 = 0, then φ(C0 ) ⊂ C1/2 ; (ii) If c0 > 1, then either φ ≡ 0 or φ(C0 ) ⊂ C0 .
The set of such Φ is called the Gordon–Hedenmalm class and denoted by G . With the same convergence and mapping properties, the Gordon–Hedenmalm theorem was extended to the P −s weighted Hilbert spaces Dα which consists of Dirichlet series F (s) = ∞ that satisfy n=1 a n n P∞ 2 α n=1 |a n | /d (n) < ∞ in [2]. Here d (n) is the divisor function which counts the number of divisors of n and α > 0. In particular, for c0 = 0, the composition operators map Dα into Dβ with β = 2α − 1. It should be noticed that Dβ are spaces that are smaller than Dα when 0 < α < 1 and bigger when α > 1. 2010 Mathematics Subject Classification. 47B33, 30B50, 11N37. The author is supported by the Research Council of Norway grant 227768. 1
We observe from the proof of this extension (see [2, Theorem 1]) that Dα are actually mapped P −s into weighted Hilbert spaces that consist of Dirichlet series F (s) = ∞ satisfying n=1 a n n ∞ X
n=1
|an |2 /(1 + Ω(n))β < ∞,
where Ω(n) is the number of prime factors of n (counting multiplicities). We say that an arithP metic function f has average order g if X1 n6 X f (n) ≍ g (X ). Since d (n)α has average order (log n)β and Ω(n)β has average order (log log n)β (see Proposition 1), Dα are in fact mapped into smaller weighted Hilbert spaces. In this paper, we show that the phenomenon of gaining one more fold of the logarithm persists for more general weights that have average order (log j n)α with j ∈ N and real α > 0. Let log j x denote the j -fold logarithm of x so that log1 x = log x and log j x = log j −1 (log x). For convenience, we define log0 x := x and ³ ´+ © ª log+ |x| := max |x|, 0 ; log+j |x| := log+ log j −1 |x|, j > 2. Define (2)
½
Hw := F (s) =
∞ X
an n
−s
¾ ∞ |a |2 X n kF kHw := 0 and integer j > 0, let ½ ∞ ∞ X X |an |2 an n −s : kF k2H = Hlog,0 := F (s) = α 1.
Theorem 2. Let α > 0 be a real number and j > 1 be an integer. When the weight {w n }n of Hw has average order (log+j n)α , a function Φ : C1/2 → C1/2 of the form defined in (1) with c0 = 0 generates a composition operator CΦ : Hw → Hlog,j −1 if and only if Φ ∈ G . There are a few things we should make clear. First, it is proved in Section 4 that the average order of (log+j Ω(n))α is (log+j +2 n)α , so that iterates of CΦ acting on Hw fit into the scope of this theorem. Second, it is natural to replace Dα with Hw and w n = d α+1 (n) . Here, when α is a positive integer, d α (n) is the number of representations of n as a product of α integers, so d 2 (n) = d (n). For general α, d α (n) is the coefficient of the nth term of the Dirichlet series of ζ(s)α , i.e. ! ! Ã Ã kr + α − 1 k1 + α − 1 k k , for n = p 1 1 · · · p r r . ··· d α (n) = kr k1 It can be checked that the proof of Theorem 1 of [2] carries through, so that CΦ maps Hw with w n = d α+1 (n) into HΩ . Notice that d α+1 (n) has average order (log n)α [11] and d (n)α has averα age order (log n)2 −1 . 2
2. P RELIMINARIES In [7], H 2 was identified with a space of analytic functions on D∞ ∩ ℓ2 (N), where D∞ is the infinite polydisk © ª D∞ := z = (z 1 , z 2 , . . . ), |z j | < 1 .
This is obtained by using the Bohr lift of Dirichlet series that are analytic in C1/2 , which is defined in the following way. Let ∞ X (3) F (s) = an n −s , ℜs > 1/2. n=1
We write n as a product of its prime factors
k
k
n = p11 · · · pr r ,
where the p j are the primes in ascending order. We replace p −s with z j , set κ(n) = (k 1 , . . . , k r ), j and define the formal power series ∞ X an z κ(n) B F (z) = n=1
as the Bohr lift of F . For χ = (χ1 , χ2 , . . . ) ∈ C∞ , we define a completely multiplicative function by requiring χ(n) = k k χκ(n) when n = p 1 1 · · · p r r and κ(n) = (k 1 , . . . , k r ). For Φ of the form (1) with c0 = 0, Φχ (s) = φχ (s) =
∞ X
cn χ(n)n −s .
n=1
Lemma 1. Suppose that Φ ∈ G . Then Φχ ∈ G for any χ ∈ D∞ . Proof. This was proved in [2, Lemma 8].
Lemma 2. Suppose that Φ ∈ G of the form (1) with c0 = 0. For every Dirichlet polynomial F , every χ ∈ D∞ and every s ∈ C0 , we have (4)
(F ◦ Φ)χ (s) = (F ◦ φχ )(s).
Proof. It was proved in [2, Lemma 9] that (F ◦ Φ)χ (s) = (Fχc0 ◦ Φχ )(s)
whenever Φ ∈ G . This is reduced to (4) when c0 = 0.
We shall now introduce a scale of Bergman spaces over D, as well as the corresponding Bergman spaces over C1/2 which are induced by a certain conformal mapping τ : C1/2 → D. Let e j := exp(exp(· · · exp(e))) (e 0 = 1). For α > 0 and j > 1, we define {z } | j e’s
d v α,1 (z) :=d A α (z) = α(1 − |z|2 )α−1 d A(z);
à µ µ ¶!−1 µ ¶¶−(α+1) jY −2 e j −1 α eℓ d v α,j (z) := logℓ d A(z), log j −1 1 − |z|2 ℓ=1 1 − |z|2 1 − |z|2 3
j > 1.
Let D α,j (D) be the set of functions f that satisfy Z k f k2D α, j (D) := | f (z)|2 d v α,j (z) < ∞. D
For f (z) =
P∞
n=0 c n z
n
, we have
k f k2D α, j (D) ≍
∞ X
n=0
Let (5)
τ(s) =
³
|cn |2
1 + log+j −1 n
´α .
s − 3/2 s + 1/2
which maps C1/2 to D. The measure µ j (s) on C1/2 induced by τ is d µ1 (s) =4α α (σ − 1/2)α−1
d m(s) ; |s + 1/2|2α+2
à !−1 à !−(α+1) jY −2 2 e j −1 |s + 1/2|2 d m(s) α + e ℓ |s + 1/2| + logℓ log j −1 , d µ j (s) = (σ − 1/2) ℓ=1 2(2σ − 1) 2(2σ − 1) |s + 1/2|2
j > 1.
Finally, let D α,j ,i (C1/2 ) consist of functions F that are analytic in C1/2 such that Z 2 |F (s)|2 d µ j (s) < ∞ (6) kF kD α, j ,i (C1/2 ) := C1/2
and D α,1,i (C1/2 ) =: D α,i (C1/2 ).
Then kF k2D α, j ,i (C1/2 ) and
=
Z
D
|F ◦ τ−1 (z)|2 d v α,j (z) = kF ◦ τ−1 k2D α, j (D) D α,1 (D) =: D α (D).
The proof of the main theorem will be given in Section 3. We verify that the average order of (log+j Ω(n))α is (log+j +2 n)α in Section 4. 3. P ROOF OF T HEOREM 2 As in [2, Subsection 3.1], we inherit the proof of the arithmetical condition of c0 from [5, Theorem A]. For the mapping and convergence properties of φ, we follow Subsection 3.2 in [2] as well. Lemma 3. Assume that w n > 1. There exists a function F ∈ Hw such that
(1) For almost all χ ∈ T∞ , Fχ converges in C0 and cannot be analytically continued to any larger domain; (2) For at least one χ ∈ T∞ , Fχ converges in C1/2 and cannot be analytically continued to any larger domain. 4
Proof. It was shown in [5] that the function F (s) =
X
p prime
p
1 p −s p log p
satisfies conditions (1) and (2). Clearly, F is in Hw because w n > 1.
The rest of the proof consists of two steps. We shall first embed Hw into certain Bergman spaces, and then apply Littlewood’s subordination principle to functions in these Bergman spaces. Lemma 4 (Embedding of Hw ). Let the weight {w n } of Hw have average order (log j n)α . Then Hw is continuously embedded into D α,j ,i (C1/2 ). For every τ ∈ Z, we define Q τ = (1/2, 1] × [τ, τ + 1). It suffices to prove the following local embedding for Hw , Z sup |F (σ + i t )|2 d t d µ∗j (σ) ≪ kF k2Hw . τ∈R Q τ
The case when j = 1 was established in [9]. We shall use the same method to establish the general case. It will suffice to prove the inequality Z1 Z1 |F (σ + i t )|2 d t d µ∗j (σ) ≪ kF k2Hw , (7) 1/2 0
where
à !−1 µ ¶−(α+1) jY −2 e j −1 e α ℓ + + ∗ log log j −1 d σ, d µ j (σ) := (σ − 1/2) ℓ=1 ℓ (σ − 1/2) (σ − 1/2)
j > 1.
We need the following lemma. Lemma 5. For α > 0 and j > 2, letting n → ∞, we have !−1 Ã µ³ Z1 jY −2 ³ ´−α−1 ¶ e j −1 ´−(α+1) d t e 1 ³ + ´−α ℓ + + + −t + O log j n . log j −1 log j n logl = n t t t α 0 l =1
Proof. We first prove the case j = 2 which is given by the integral Z1 ³ e ´−(α+1) d t I := e −t log n log . t t 0
We split the integral at the point t = 1/ log n, which is dictated by the exponential decay of the integrand. This gives I = I 1 + I 2, where
I1 =
Z1/ log n
³ e ´−(α+1) d t e −t log n log t t
I2 =
Z1
−t log n
and
0
e
1/ log n 5
³
e ´−(α+1) d t . log t t
For I 2 , we split it again at the point t = p 1
log n
Z1/plog n
e −t log n
1/ log n
Z1 1/
p
e
³
Z1/plog n e ´−(α+1) d t dt 1 log e −t log n ≪¡ p ¢ 1+α t t t 1/ log n loge log n Z∞ dt 1 e −t ≪¡ ¢α+1 t 1 log2 n 1 ≪¡ ¢α+1 ; log2 n
−t log n
log n
:
³
e ´−(α+1) d t q ≪ log n log t t
Z1 1/
p
e −t log n d t
log n
p e − log n ≪p . log n
¡ ¢ For I 1 , we write e −t log n = 1 + O t log n Z1/ log n
¡ ¡ ¢¢ ³ e ´−(α+1) d t 1 + O t log n log t t 0 ¶ µ Z∞ Z1/ log n ³ ¡ ¢−(α+1) d t e ´−(α+1) = dt log(e t ) + O log n log t t log n 0 ³¡ ¢−α ¢−α−1 ´ 1¡ = log2 n + O log2 n . α
I1 =
For the general integral with j > 2 we can follow the same procedure. The main contribution comes from the term I 1 , and I 2 gives a negligible contribution, that is µ³ ´−α−1 ¶ 1 ³ + ´−α + I 1 = log j n , + O log j n α p ³ ´−α−1 e − log n . + p I 2 ≪ log+j n log n
Proof of Lemma 4. Let F ∈ Hw . Using duality, we have 6
Z1 0
¯ ¯Z ¯ ¯ 1X ¯ ¯ −σ−i t |F (σ + i t )| d t = sup ¯ an n g (t )d t ¯ ¯ ¯ 0 n >1 g ∈L 2 (0,1) 2
kg k2 =1
6
¶
µ
∞ |a |2 X n n −2σ+1 (log+j n)α w n n=1 {z } | (i)
¯ ¯2 ∞ ¯gˆ (logn)¯ w X n sup , + α n(log j n) g ∈L 2 (0,1) n=1
|
kg k2 =1
{z
}
(ii)
where gˆ is the Fourier transform of g . By the smoothness of gˆ and the assumption on w n , the supremum on the right hand side is finite. Integrating both sides against d µ∗j (σ) and applying Lemma 5, we get the inequality (7).
Lemma 6. For ω : D → D and f ∈ D α,j (D), there exists some constant C 0 depending on ω(0) such that k f ◦ ωk2D α, j (D) 6 C 0 k f k2D α, j (D) , i.e., Z
(8)
D
¯ ¯ ¯ f (ω (z))¯2 d v α,j (z) 6 C 0
Z
D
¯ ¯ ¯ f (z)¯2 d v α,j (z).
Proof. Suppose ω(0) = a, and let ψ : D → D be analytic such that ψ(z) = ωa ◦ ω(z) with ωa (z) = z−a ¯ . Then we have ω(z) = ωa ◦ ψ(z). Starting with Littlewood’s subordination principle, 1−az Z2π ¯ ³ ³ ´´¯2 d θ Z2π ¯ ¯ ¯ ¯ = ¯f ω r eiθ ¯ ¯f 2π 0 0 Z2π ¯ ¯ 6 ¯f 0
since ψ(0) = 0. Therefore, Z ¯ ¯ ¯ f (ω(z))¯2 d v α,j (z)
³ ´´¯2 d θ ¯ ωa ◦ ψ r e i θ ¯ 2π ³ ³ ´´¯2 d θ ¯ ωa r e i θ ¯ 2π ³
D
¡
2
6 1 − |a|
6 Ca
µ
¢
Z
D
1 + |a| 1 − |a|
à −1 ¯ ¯2 jY ¯ f (z)¯ log+ ℓ ℓ=1
¶Z
D
eℓ 1 − |ωa (z)|2
!−1 µ
log+j
ej 1 − |ωa
(z)|2
¶−(α+1)
¡
d A(z) ¢ ¯ 2 1 − |z|2 |1 − az|
¯ ¯ ¯ f (z)¯2 d v α,j (z)
for some C a depending on a.
7
¡ ¢ Proof of Theorem 2. When c0 = 0, by Lemma 2, (F ◦ Φ)χ (s) = F ◦ Φχ (s) for every Dirichlet polynomial F , χ ∈ D∞ and s ∈ C0 . For fixed s, λ ∈ D, χ ∈ T∞ and λχ := (λχ1 , λχ2 , λχ3 , . . . ), we may view ∞ X cn λΩ(n) χ(n)n −s = φλχ (s) := η s (λ) Φλχ (s) = n=1
as an analytic map η s : D → C1/2 with η s (0) = c1 . Putting ω = τ ◦ η s = τ ◦ Φλχ and applying τ to the inequality (8) with f = F ◦ τ−1 and being a Dirichlet polynomial, we have
1 + τ(c1 ) kF ◦ Φλχ k2D α, j (D) 6 C 1 kF k2D α, j ,i (C1/2 ) . 1 − τ(c1 ) P −s As in [2], we assume F ◦Φ(s) = ∞ n=2 b n n . To avoid negative arguments in the j -fold logarithm, ∞ we shall equip χ ∈ T with an indicator function with respect to the value of Ω(n) by defining ª (n). χ (n) = χ(n) · 1© (9)
j
P∞
n: Ω(n)>e j −3
−s
Ω(n)
χ j (n)n into (9) and integrate against the Haar measure Then we put F ◦ Φλχ j = n=2 b n λ ∞ d ρ(χ) over T to get ¯2 Z Z ¯X X ¯ ¯∞ |b n |2 n −2σ Ω(n) −s ¯ ¯ d v (λ)d ρ(χ) ≍ b λ χ (n)n (10) ³ ´α . α,j n j ¯ ¯ ª © T∞ D n=1 1 + log Ω(n) n: Ω(n)>e j −3 j −1 Upon letting σ → 0 we have
|b n |2
1 + τ(c1 ) kF k2D α, j ,i (C1/2 ) ´α 6 C c1 ª © 1 − τ(c ) 1 1 + log j −1 Ω(n) n: Ω(n)>e j −3 X
³
for some constant depending on c1 . We get our conclusion by Lemma 4.
Even though we may get C 0 = 1 in Lemma 6 by requiring ω(0) = 0 = τ◦η s (0) = τ (c1 ), we cannot claim the contractivity due to the constant appearing in the embedding. 4. T HE AVERAGE ORDER In this section, we will verify that the average order of (log+j Ω(n))α is (log+j +2 n)α . It will suffice to give the details when j = 0. Proposition 1. For real α > 1, we have h ¡ X ¡ ¢α ¢α−1 i Ω(n)α = X log2 X + O X log2 X (11) n6X
This estimation is consistent with the case when α = 1 or 2 which can be found in [12]. Let © ª Nk (X ) := # n 6 X : Ω(n) = k
and
α SΩ (X ) =
X
Ω(n)α .
n6X
α α (X ) as We shall use some results of Nk (X ) to estimate S Ω (X ), for which we need to rewrite S Ω X α SΩ (X ) = k α Nk (X ). log X
16k 6 log2 8
Proof of Proposition 1 . The quantity Nk (X ) has several changes in its behaviour as k varies with X . These are described in [12] (see Theorems 5 and 6 in Chapter II.6) and [8]. Accordingly, we α split the sum S Ω (X ) into different parts with respect to k. These are given by: (i) (ii) (iii) (iv)
1 6 k 6 (1 − ǫ) log2 X ; (1 − ǫ) log2 X < k < (1 + ǫ) log2 X ; (1 + ǫ) log2 X 6 k < A log2 X ; log X A log2 X 6 k 6 log 2 . p Here A > 3 and ǫ = 2B log3 X / log2 X with B some large constant. With this choice of ǫ the sum over the range (ii) is centred about the mean of Ω(n) and hence should give the main contribution. We first concentrate on this range. Theorem 5 in Chapter II.6 of [12] states that ½ µ ¶ µ ¶¾ k −1 k X (log2 X )k−1 ν +O , Nk (X ) = log X (k − 1)! log2 X (log2 X )2 where
µ ¶ µ ¶ Y 1 z −1 1 z ν(z) := 1− 1− , Γ(z + 1) p p p
Therefore, (1+ǫ)X log2 X
k=(1−ǫ) log 2
(1+ǫ)X log2 X
|z| < 2.
µ µ ¶ µ ¶¶ k −1 1 k ν +O (k − 1)! log2 X log2 X k=(1−ǫ) log 2 X ¡ ¢k µ µ ¶ µ ¶¶ (1+ǫ)X log2 X k −1 1 X α+1 log2 X k ν +O . = log X log2 X k=(1−ǫ) log X k! log2 X log2 X
X k Nk (X ) = log X X α
k−1 α (log2 X )
2
Applying Stirling’s formula gives the sum µ
µ
1 1+O p log2 X 2π log X log2 X X
¶¶
(1+ǫ)X log2 X
k
k=(1−ǫ) log 2 X
α+1/2
¡
log2 X kk
¢k
ek
¶ µ ¶¶ µ µ 1 k −1 +O . ν log2 X log2 X
We now write k = log2 X + ℓ with
so that
q ´ ¡ ¢ ³ q ℓ ∈ −ǫlog2 X , ǫlog2 X = − 2B log2 X log3 X , 2B log2 X log3 X ¶ ¶ µ ¶ µ ℓ−1 1 ′′ 1 ℓ−1 2 k −1 ′ =ν(1) + ν (1) +O + ν (1) ν log2 X log2 X 2 log2 X log2 X ¶2 µ ¶ µ ℓ 1 ′′ 1 ℓ ′ =1 + ν (1) +O + ν (1) log2 X 2 log2 X log2 X µ
and (12)
k α+1/2 = (log2 X )α+1/2
µ
ℓ 1+ log2 X
¶α+1/2
! Ã ¶m ∞ α + 1/2 µ X ℓ α+1/2 = (log2 X ) . m log2 X m=0
9
Upon forming the product of these two series, our sum becomes ¢log2 X +ℓ ℓ µ µ ¶¶ ǫ log2 X ¡ X log2 X e X (log2 X )α−1/2 1 1+O p log X log2 X ℓ=−ǫ log X (log2 X + ℓ) 2 +ℓ 2π 2 µ ¶2 µ ¶¶ µ ℓ 1 ℓ × 1 + c1 +O . + c2 log2 X log2 X log2 X for some coefficients c j . We also expand the first factor of our sum as follows ¡ ¢log2 X +ℓ ℓ log2 X e =e ℓ−(log2 X +ℓ) log(1+ℓ/ log2 X ) (log2 X + ℓ)log2 X +ℓ !m # " à n ∞ ∞ 1 X X ℓ2 ℓ − 2 log 2X . 1+ =e ¡ ¢n−1 m=1 m! n=3 n(n − 1) log2 X
Thus, our summand takes the form ¶2 µ h ℓ2 ℓ ℓ3 ℓ − 2 log X 2 (13) e 1 + c1 + + c2 log2 X log2 X 6(log2 X )2 ¶µ ¶4 µ ¶i µ 1 ℓ 1 c1 +O + log2 X + . 6 12 log2 X log2 X Upon performing the sum we only retain the terms with an even power of ℓ. Hence, we are led to compute µ ¶¶ ǫ log2 X ¶2 µ µ h X ℓ2 X (log2 X )α−1/2 ℓ 1 − 2 log X 2 1 + c2 e + 1+O p log2 X ℓ=−ǫ log X log2 X 2π 2 µ ¶µ ¶4 µ ¶i c1 1 ℓ 1 + +O log2 X + . 6 12 log2 X log2 X On approximating the sum µwith an integral via the Euler–Maclaurin summation formula we ¶
gain an error term of order O Zp2B log p
−
since
2X
log3 X
1
(log2 X )B
! Z∞ 2 2 −u 2X dt = e du 2π log2 X 1 − p p B log3 X 2𠶶 µ µ q 1 = 2π log2 X 1 + O (log2 X )B
2
e
t − 2 log
2B log2 X log3 X
Z∞
. Then the leading term is given by Ã
q
Z 2 1 ∞ −u2 1 e ue du 6 d u ≪ e −a . a a a a For the higher powers of ℓ we use the formulae ¶ µ Z∞ 1 −u 2 2n e u du = Γ n + 2 ∞ −u 2
10
and
Z∞ a
2
e −u u 2n d u ≪ a 2n−1 e −a
2
which follow from Z∞ Z Z 1 ∞ 2n+1 −u2 1 2n−1 −a 2 n ∞ 2n−1 −u2 2n −u 2 e + u e du 6 u e u e du = a d u. a a 2 a a a These give p Zp2B log X log X ³q ´2n+1 Z B log3 X t2 3 2 2 − 2 log 2n 2X t dt = e e −u u 2n d u 2 log2 X p p −
2B log3 X log2 X
=
³q
Putting everything together gives (1+ǫ)X log2 X
2 log2 X
´2n+1
B log3 X
−
¡ ¢ Γ(n + 1/2) + O (log2 X )n+1/2−B (log3 X )n−1/2
³ ¡ ¡ ¢α ¢α−1 ´ k Nk (X ) = X log2 X + O X log2 X . α
k=(1−ǫ) log 2 X
It should be clear from the above that with more precision one can obtain an asymptotic expansion to any required degree of accuracy. For (i) and (iii), we shall use the Erd˝os–Kac theorem for Ω(n)[11], which states that ) ( Zb t2 Ω(n) − log2 n X 6b ∼ p e− 2 d t . (14) # n6X :a6 p log2 n 2π a This gives
(1−ǫ)X log2 X k=1
log2 X ¡ ¢α (1−ǫ)X Nk (X ) k α Nk (X ) ≪ log2 X k=1
¡
= log2 X
¢α
X
1.
n6X 16Ω(n)6(1−ǫ) log 2 X
|
{z
=:S 1,X
}
Similarly, A log X2 X
k=(1+ǫ) log 2 X
¡ ¢α k α Nk (X ) ≪ log2 X
X
|
{z
=:S 3,X
For X large, there exits n 0,X such that for n > n 0,X q q Ω(n) − log2 n 6 C 2 ǫ log2 X −C 1 log2 X 6 p log2 n
and
q q Ω(n) − log2 n C 1′ ǫ log2 X 6 p 6 C 2′ A log2 X , log2 n 11
1.
n6X (1+ǫ) log2 X 6Ω(n)6 A log 2 X
}
for some C 1 , C 2 , C 1′ , C 2′ . Therefore, S 1,X and
X ∼p 2π
ZC 2 A plog C1ǫ
p
2X
log2 X
for some ǫ0 , ǫ′0 > 0, respectively.
t2
e− 2 d t ≪ X
q
S 3,X ≪ ¡
t2 ¯ log2 X e − 2 ¯t=C ǫplog 1
X
log X
2X
≪¡
X log X
¢ ǫ0
¢ ǫ′
0
For (iv), by Nicolas’s result in [8], there exists the same constant C , such that ½µ ¶ µ ¶¾ log XX / log 2 log XX / log 2 CX X α α k k Nk (X ) ≪ log 2k 2k k=A log2 X k=A log2 X µ ¶ log XX / log 2 X α −k k 2 log k . =C X 2 k=A log 2 X
We may bound this last sum from above by the integral µ ¶ Zlog X / log 2 Z∞ X α −t t 2 log t d t 6 log X t α 2−t d t 2 A log2 X A log2 X ¡ ¢α 6 log X A log2 X 2−A log2 X ¡ ¢α ¡ ¢−A log 2 ≪ log X log2 X log X ¡ ¢1−3log 2 ¡ ¢α 6 log X log2 X .
¡
¢α For the average order of log+j Ω(n) , when carrying through the proof of Proposition 1 for ³ ´α the range (ii), it can be seen from (12) that the main contribution log j +2 X gets more and more centralised when j becomes bigger. Therefore, we eventually get ³ ´α ³ ´ log X X ¡ + α ¢α j +2 log j Ω(n) = X log j +2 X + O X . log X n6X 2 ACKNOWLEDGEMENTS
I thank Ole Fredrik Brevig for introducing me to this topic and giving valuable information during my work. I also thank Winston Heap and Kristian Seip for crucial advice on the proof of the asymptotic approximation. I thank all of them for their kind suggestions on early drafts of this paper. R EFERENCES [1] M. Bailleul, Composition operators on weighted Bergman spaces of Dirichlet series J. Math. Anal. Appl. 426 (2015), no. 1, 340-363. [2] M. Bailleul, O. F. Brevig, The composition operators on Bohr-Bergman spaces of Dirichlet series, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 1, 129-142. 12
[3] F. Bayart, Hardy spaces of Dirichlet Series and their composition operators, Monatsh. Math. 136 (2002), no. 3, 203-236. [4] F. Bayart, O. F. Brevig, Composition operators and embedding theorems for some function spaces of Dirichlet series, arXiv:1602.03446. [5] J. Gordon, H. Hedenmalm, The composition operators on the space of Dirichlet series with square summable coefficients, Michigan Math. J. 46 (1999), no. 2, 313-329. [6] G. H. Hardy, S. Ramanujan, The normal number of prime factors of a number n, Quart. J. Math. 48 (1917), 76-92. [7] H. Hedenmalm, P. Lindqvist and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L 2 (0, 1), Duke Math. J. 86 (1997), no. 1, 1-37. [8] J.–L. Nicolas, Sur la distribution des nombres entiers ayant une quantité fixée de facteurs premiers, Acta Arith. 44 (1984), no. 3, 191-200. [9] J.–F. Olsen, Local properties of Hilbert spaces of Dirichlet series, J. Funct. Anal. 261 (2011), no. 9, 26692696. [10] H. Queffélec, K. Seip, Approximation numbers of composition operators on the H 2 space of Dirichlet series, J. Funct. Anal. 268 (2015), no. 6, 1612-1648. [11] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press. 46 (1995). [12] P. Turán, On a theorem of Hardy and Ramanujan, J. London Math. Soc. S1-9 (1934), no. 4, 274. [13] A. Zygmund, Trigonometric series, Cambridge University Press. (2002). J ING Z HAO , D EPARTMENT OF M ATHEMATICAL S CIENCES , N ORWEGIAN U NIVERSITY OF S CIENCE AND T ECHNOL NO-7491 T RONDHEIM , N ORWAY E-mail address:
[email protected];
[email protected]
OGY,
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