On counting centralizer subgroups of symmetric groups

Let $S_{2m}$ be the symmetric group, $h=(1\ 2)(3\ 4)\cdots(2m-1\ 2m)$ and $H=C(h)$. We consider the structure of $gHg^{-1}\cap H$ for any $g\in S_{2m}$. We prove the permutations $g$ which makes $gHg^{-1}\cap H$ have size of polynomial in $m$ have density zero.


Introduction
The paper is motivated by a MathOverflow question of Harald Helfgott [1], which relates to growth in transitive permutation groups in his work [6]. For any positive integer m, let S 2m be the symmetric group on the symbols {1, 2, · · · , 2m} and H be the subgroup of S 2m consisting of permutations preserving the partition {1, 2}, {3, 4}, · · · , {2m − 1, 2m}, or equivalently H = C((1 2)(3 4) · · · (2m − 1 2m)), the centralizer subgroup. Call g ∈ S 2m good if |H ∩ gHg −1 | = m O(1) ; call it bad otherwise. Then Helfgott wonders about the structure of good elements and postulated that the good permutations have density 1 in S 2m . There seems to be a fair share of good permutations in S 2m , for example if the cycle decomposition of g does not contain "too many" cycles of the same length, g may be checked good. The paper contributes to studying the structure of good elements, and shows that the good permutations have density zero as a negative answer to Helfgott's postulation.
To proceed, we first clarify the structure of H ∩ gHg −1 for any g ∈ S 2m in section 1. It turns out that the isomorphism class of H ∩ gHg −1 depends on the double coset HgH and moreover Theorem 1. Each HgH has a representative x ∈ Sym{2, 4, · · · , 2m} ≤ S 2m determined by a partition of m and there is a one-to-one correspondence H\S 2m /H ↔ {parititions of m}. Furthermore, for any g ∈ HxH with x ∈ Sym{2, 4, · · · , 2m} whose cycle decomposition has r i cycles of length i, i = 1, · · · , k, where D i is the dihedral group with 2i elements. (For convenience we write D 1 for C 2 or S 2 .) Thus |H ∩ gHg −1 | can be seen as a random variable on partitions of m with probability distribution of counting measure P (λ) = |HxH| |S 2m | , if g ∈ HxH for x ∈ Sym{2, 4, · · · , 2m} with cycle type λ. Then we prove Theorem 2. For any c > 0, P (|H ∩ gHg −1 | < m c ) → 0, as m → ∞. 1 Consequently, good elements of S 2m have density zero.
The right tail of P is also estimated to show that Theorem 3. For some constant C > 0, P |H ∩ gHg −1 | > Cm log m → 0, as m → ∞.
In particular, the bad elements g ∈ S 2m with |H ∩ gHg −1 | ≫ m log m have zero density.
Outline of paper. The 1-1 correspondence H\S 2m /H ↔ {parititions of m} of Theorem 1 is established inductively by studying the left and right action of H on S 2m in details in section 1. 3. It can also be verified by a character formula in section 1.4. Then combined with an idea of bipartite graph automorphism construction introduced by J. P. James [11], we prove the structure result in Theorem 1 in section 1. 5. As a byproduct we prove that they are all rational groups in aspect of representation theory. Section 1. 6 gives some computational verification of Theorem 1.
Explicitly shown in section 2.1, the distribution of |H ∩ gHg −1 | happens to be P = ESF( 1 2 ), where ESF( 1 2 ) is the Ewens' distribution with bias 1 2 . Then we estimate the left tail P (≤ m c ) by the moment bound. The expectations for each m involved in the moment bound are assembled into a special generating function. Then asymptotics of the expectations can be extracted from coefficients of singular expressions of the generating function around its singularities which are of logarithmic type, see section 2.2.2. We use techniques from analytic combinatorics, especially the hybrid method introduced by Flajolet et al [4], to find the correct asymptotics and prove Theorem 2 in section 2.3. Following the same probabilistic setting, the expectations involved in the moment bound of the right tail are assembled in to generating functions with singularities of exponential type. Then to prove Theorem 3, we use asymptotics of coefficients of generating functions of exponential type which was given by E. M. Wright [12], in section 3. 1.2. Double coset decomposition of S 2m . Counting the left cosets contained in HgH gives Thus if each double coset determines a distinct structure (or size) of H ∩ gHg −1 , the density of those g is assigned by In addition, the double coset decomposition of S 2m by H gives and consequently by Stirling's formula. These formulas become the starting point of studying distribution of |H ∩ gHg −1 | in section 2.
1.3. Counting double cosets by partition number. To describe the structure of H\S 2m /H, we first prove the following lemma on double coset representatives.
Proof. We use induction on m. m = 1 is trivial. Suppose the lemma is true for any m ′ < m. For any x ∈ S 2m , if x = yz with y and z supported on the 2-blocks t∈N {t, t + 1} and t / ∈N {t, t + 1} respectively for some proper subset N ⊂ M (in particular y commutes with z), then by induction y and z can be made into permutations on N and M N through multiplying on left and right by H restricted to t∈N {t, t + 1} and t / ∈N {t, t + 1} respectively. Hence HxH has a representative supported on M .
In addition, the following explicit expression of Proposition 4 is crucial to proving the main result of this section.
Sym{2i − 1, 2i} ≤ S 2m , and T = Sym{(1, 2), · · · , (2m − 1, 2m)} ≤ S 2m (the symmetric group of the ordered pairs (2k − 1, 2k)'s). Then H = T C and explicitly for any h ∈ H, there is a unique decomposition in whichh ∈ C, h ∈ T , h M and h M ′ , commuting with each other, are the complementary permutation actions of h restricted onto M and M ′ respectively. We call it the TC-decomposition of H.
Proof. For any h ∈ H and k ≤ m, let h be the permutation action defined as where h · i denotes the number that h moves i to.
The definition guarantees that h sends even numbers to even numbers and odd to odd while still preserving the partition {1, 2}, · · · , {2m − 1, 2m}, hence belongs to H. The case separation in the definition where 2k − 1 and 2k are switched by h gives a product of transpositions (2k −1, 2k)'s, denoted byh. This amounts to the decomposition h = hh which is unique simply because C ∩ T = {1}. Restriction of h onto M and M ′ gives the 3-term decomposition whose uniqueness is due to the decomposition T = SymM × Sym(M ′ ).
On the other hand, if x 2 ∈ Hx 1 H, then there exists h ∈ H such that x 1 hx −1 2 ∈ H. By Lemma 7 we get (1) It is easy to check that chc −1 = chc ∈ Sym(M ) (Sym(M ′ )) for any h ∈ Sym(M ) (Sym(M ′ )) and any c ∈ C such that h preserves the support of c, denoted by supp(c). We claim that 2h ∈ Sym(M ). Therefore in (1), we can switch h M ′ and (hx −1 2h ) to get which must be the 3-term TC-decomposition of h. Hence x 1 hx −1 2 = h and Furthermore we can choose h ∈ H withh = 1, so that x 1 is conjugate to x 2 by h M ∈ Sym(M ). Actually since x −1 2 preserves supp(h), it is easy to verify that x 2h x −1 2 ∈ H. Then the TC-decomposition Thus we can choose h ∈ T in the beginning. Finally, since Sym(M ) ≃ S m in an obvious way, the conjugacy classes of Sym(M ) hence the double cosets H\S 2m /H are in one-to-one correspondence with the partitions of m.
Remark 10. Note that if x preserves the support of c ∈ C, then cxc is the truncation of x from supp(c), i.e. cxc | supp(c) is the trivial permutation and cxc is the same permutation as x outside of supp(c).
Remark 11. Now we can show by Stirling's formula that Since p(m) ∼ 1 4 √ 3m e π √ 2m/3 by Hardy-Ramanujan [7], the average is of super-polynomial growth, which is a sign that the density of good elements should be low.

1.4.
Counting double cosets by character formula. Apart from the combinatorial methods in section 1.3, there is also an applicable method of counting (self-inverse) double cosets by character formula.
Proposition 12 (J.S. Frame [5], Theorem A). The number of self-inverse double cosets of a finite group G with respect to a subgroup H ≤ G equals where the sum is over Frobenius-Schur indicators of irreducible characters occurring in the induced character of G from the trivial character of H. Here for any character χ of G, Note that Ind G H 1 H is afforded by the permutation representation of G through its action on the right cosets H\G.
Proof. We follow the ideas of [5].
First, we show that the number of self-inverse double cosets of G with respect to H is (See Theorem 3.1 of [5].) It suffices to show that each self-inverse double coset corresponds to |G| solutions to the equation , which says that the inverse of t = g i xg −1 i belongs to its own right coset. Each double coset HgH decomposes into right cosets as hence each left coset h ′ gH ⊂ HgH intersects with each right coset Hgy at h ′ g(g −1 Hg ∩H)y, all of which have d = |g −1 Hg ∩H| elements. In particular, the inverse of each right coset is a left coset, so it intersects with its own right coset at d elements, which count as d values of t i . Summing over all right cosets in HgH, we get [H : (g −1 Hg ∩ H)]d = |H| solutions to (2) in HgH if it is an self-inverse double coset. Varying the right cosets g i ∈ H\G, , which amount to [G : H]|H| = |G| solutions. Now let G act on H\G by right multiplication and consider the corresponding permutation representation of G, which affords Ind G H 1 H by definition. Since the character value of a permutation representation on every element is the number of its fixed points, we get

Next, we resort to an interesting result of Inglis-Richardson-Saxl [8] on multiplicity free decomposition of the permutation representation Ind
where S ν for any partition ν denotes the Specht module (over Q).
By Proposition 9, the double cosets of S 2m with respect to H are all selfinverse for x conjugate to x −1 in Sym { 2, 4, · · · , 2m}. Also note that all irreducible representations of symmetric groups are of real type, i.e. F S(χ) = 1 for any χ ∈ Irr S 2m . Then Proposition 13 and Proposition 12 show that the number of double cosets H\S 2m /H equals the partition number.
1.5. Structure of H ∩ gHg −1 and proof of Theorem 1. With the structure description of double cosets H\S 2m /H, this section proves Theorem 1 using an idea of constructing bipartite graph automorphisms introduced by J.P. James [11].
We allow one edge to be duplicated. A graph automorphism is a permutation of vertices that sends edges to edges. Denote Aut b (G) the set of automorphisms preserving V i , i = 1, 2. Suppose G is k-regular, i.e. each vertex belongs to k edges, then |E| = kl for some positive integer l. Label the edges by integers between 1 and kl. Define two k-partitions of {1, · · · , kl} as , v belongs to i}, the set of all edges containing v. Then any automorphism of Aut b (G) is a permutation of {1, · · · , kl} that preserves the two k-partitions α 1 , α 2 . Denote the group of such permutations On the other hand, each permutation of (S kl ) α 1 ,α 2 is an automorphism of Aut b (G). This is simply because each part of α i (a k-subset of {1, · · · .kl}) corresponds to a vertex in V i , hence a permutation preserving α i sends a vertex to a vertex, which also sends edges to edges by definition. We summarize Lemma 2.2 and 2.3 of [11] as follows Proof of Theorem 1. The one-to-one correspondence H\S 2m /H was already established in Proposition 9. In application of Proposition 14 to our case, let k = 2, l = m, the edges be 1, 2, · · · , 2m, and the two parties of vertices be ). By Proposition 9, the structure of H ∩ gHg −1 depends only on those g supported on even (or odd) numbers and their cycle type determined by partitions of M ′ = {2, 4, · · · , 2m}. Hereinafter we denote a partition by λ = {1 r 1 · · · k r k } which means λ has r i parts equal to i and by N λ = k i=1 r i the number of parts of λ. If g ∈ HxH for x in the conjugacy class of Sym(M ′ ) with cycle type λ , then the constructed bipartite graph G has N λ connected components corresponding to parts of λ, i.e. cycles of x. For instance, the component corresponding to a part k of λ, which may be expressed as the standard cycle (2 4 · · · 2k) ∈ Sym(M ′ ), looks like

When unfolded, it becomes a 2k-gon
Denote such a bipartite graph by G k . Clearly as a proper subgroup of the automorphism group of the above 2k-gon, i.e. D 2k , Aut b (G k ) contains the automorphism group of the k-polygon with blue nodes (or equivalently the k-gon with green nodes) and dashed edges, i.e. D k . Hence Aut b (G k ) ≃ D k , the dihedral group with 2k elements. Any automorphism in Aut b (G) can also permute components of the same size, i.e. those corresponding to cycles of the same length. Thus the above construction using bipartite graphs replicates the definition of wreath product with symmetric groups. Hence for any permutation x ∈ Sym(M ′ ) with cycle type {i r }, by Proposition 14 we have the wreath product presentation In general for any g ∈ HxH and x ∈ Sym(M ′ ) of cycle type λ = {1 r 1 2 r 2 · · · k r k }, we get and in particular, This completes the proof.
Using Theorem 1 we can measure the double cosets as follows Corollary 15. For any g ∈ HxH with x ∈ Sym{2, 4, · · · , 2m} with x of cycle type λ = {1 r 1 · · · k r k }, By Theorem 4.4.8 of James-Kerber [10], the wreath product of a rational finite group with any symmetric group is also rational, hence Theorem 1 implies Corollary 16. All irreducible representations of H ∩ gHg −1 are realizable over Q.
For the simplest example, if x ∼ {1 m }, then Theorem 1 gives where D i denotes the dihedral group with 2i elements and for convenience, we write C 2 as D 1 . Note that (4 5 GAP gives the following structure description in coincidence with Theorem 1 2 3)(4 5)(6 7)  GAP gives the following structure description in coincidence with Theorem 1 4 5)(6 7)(8 9)H(4 5)(6 7)(8 9 More computational verification by GAP for m ≥ 6 can also be checked.

Count good elements
With the structural results on H ∩ gHg −1 , we are prepared to count good elements in S 2m . Recall that g ∈ S 2m is good if |H ∩ gHg −1 | = O(m c ) for some universal constant c > 0.
2.1. Counting with random permutation statistics. We show that the distribution of |H ∩ gHg −1 | happens to be the Ewens' distribution with bias θ = 1 2 . By definition (see Example 2.19 of Arratia-Barbour-Tavaré [2]), the Ewens' distribution ESF(θ) is the distribution equipped with the following probability density on partitions λ = {1 r 1 · · · k r k } of m By Theorem 1 and Corollary 15, the distribution of |H ∩gHg −1 | over g ∈ S 2m is equivalent to the following probability density on partitions of m, i.e. for any x ∈ Sym{2, 4, · · · , 2m} of cycle type λ, which is exactly P 1 2 (λ) as in (3).
This turns the study of distribution of |H ∩ gHg −1 | into study of Ewens' distribution ESF( 1 2 ). By Theorem 5.1 of [2], as m → ∞, ESF(θ) pointwise converges to the joint distribution of independent Poisson distributions (Z 1 , Z 2 , · · · ) on N ∞ , where Z i ∼ P o(θ/i) for any i ≥ 1 with Prob(Z i = j) = e −θ/i (θ/i) j j! . However, the unmanageable errors appearing in [2] between Ewens' distributions and joint Poisson distribution make it inaccessible to calculate the tail distribution of ESF(θ). In the next section, we use methods of analytic combinatorics to estimate the left tail P (|H ∩ gHg −1 | ≤ m c ), i.e. the probability of good elements.
For any a ∈ R, define W a,m := |λ|=m f (λ) −a . Especially for a = 0 we get the partition number W 0,m = p(m) ∼ 1 4 √ 3m e π √ 2m/3 and for a = 1, by section 1.2. Also note that W a,m strictly decreases as a increases. In this notation we can write the distribution P defined in (4) as P (λ) = W −1 1,m f (λ) −1 . To estimate P (f (λ) ≤ m c ), i.e. the probability of good elements, we introduce the moment bound. For any nonnegative random variable X from a sample space Ω to R ≥0 with probability distribution F , define the α-th moment for any α > 0 by Then by Markov's inequality, we have for any C > 0, Since α is arbitrary, we get Proposition 17 (Moment bound). For any α > 0 and nonnegative random variable X with distribution F , Now for the distribution P defined in (4), the moment bound applied to X = f −1 gives for any c > 0, since we have the expectation Hence the task is to find appropriate estimate of W α+1,m for α > 0. This is accessible through a hybrid method introduced by Flajolet et al [4] which we present in section 2.3.

2.2.1.
Generating function of W β,m . Before applying the hybrid method, it is necessary to introduce the following generating function for any β ∈ R, where I β (z) = j≥0 z j (j!) β defines an entire function of exponential-like. For β > 0, W β is an analytic function in the open unit disk of convergence radius ≥ 1 at the origin, since To further determine the convergence radius of W β (z), β > 0, we need a lower bound for W β,m . For any α ∈ R, let µ α be the distribution on {partitions of m} with µ α (λ) = W −1 α,m f (λ) −α for any partition λ of m. For example, µ 0 is the uniform distribution and µ 1 is the distribution P = P 1 2 in the notation of Ewen's distribution defined in (3). For 0 < γ < 1, x 1/γ is a convex function, hence by Jensen's inequality (with expectation E µ β over µ β ), for any α, β ∈ R,
Lemma 22 (Lemma 5 of [4]). For any γ ∈ C, the polylogarithm Li γ (z) is analytically continuable to the slit plane C R ≥1 . Moreover, the singular expansion of Li γ (z) near the singularity z = 1 for non-integer γ is is the gamma function and ζ(z) is the Riemann zeta function. For m ∈ Z + , where H k is the harmonic number 1 + 1/2 + · · · + 1/k. In (10) (similar to (11)), the first term is the singular part for γ with real part Reγ ≤ 1 and the regular remainder tends to ζ(γ) = Li γ (1) if Reγ > 1, as τ → 0 (or z → 1). The lemma indicates that for 0 < β < 1, Tauberian theorem is also not directly applicable to W β (z), since e a(− log z) β−1 ≫ (1 − |z|) −a for any a > 0, i.e. is of infinite global order. In section 3, we will introduce asymptotics of coefficients of this type through a saddle point method handled by E .M. Wright [12].
Note that Li γ (z k ) only has singularities at k-th roots ξ k of unity, the above lemma gives the corresponding singular expansion (12) Li which becomes a series of (1 − z/ξ k ) by substitution

2.3.
Proof of Theorem 2 by hybrid method asymptotics for W β,m . We first introduce some necessary notions following Flajolet et al [4].
Definition 23. The global order of an analytic function f (z) in the open unit disc, is a number a ≤ 0 such that |f (z)| = O((1 − |z|) a ), ∀|z| < 1.
Since for any β > 1, W β (z) is bounded in the unit disc, its global order is zero. It can be shown by Cauchy's integral formula that a function f (z) of global order a ≤ 0 has coefficients satisfying [z n ]f (z) = O(n −a ), see section 1.1 of [4].
Definition 24. A log-power function at 1 is a finite sum of the form where α 1 < · · · < α k and each c k is a polynomial. A log-power function at a finite set of points Z = {ζ 1 , · · · , ζ m }, is a finite sum where σ j is a log-power function at 1. (1 − z), a log-power function can be seen as approximation by combinations of these two polylogarithms. Asymptotics of coefficient of log-power functions are known, see Lemma 1 of [4].
Definition 25. Let h(z) be analytic in |z| < 1 and s be a nonnegative integer. h(z) is said to be C s -smooth on the unit disc, or of class C s , if for all k = 0, · · · , s, its k-th derivative h (k) (z) defined for |z| < 1 admits a continuous extension to |z| = 1.
The smoothness condition relates to the coefficients of a function in an obvious way: if h(z) = n≥0 h n z n with h n = O(n −s−1−δ ) for some δ > 0 and s ∈ Z ≥0 , then it is C s -smooth. Conversely, we have the Darboux's transfer (Lemma 2 of [4]): if h(z) is C s -smooth, then h n = o(n −s ). By (9) and the easy differentiation formula Li ′ γ (z) = Li γ−1 (z)/z, we can see that for any β ≥ 2, W β (z) is at least C ⌊β⌋−2 -smooth on the unit disc.
Definition 26. An analytic function Q(z) in the open unit disc is said to admit a log-power expansion of class C t if there exist a finite set of points Z = {ζ 1 , · · · , ζ m } on the unit circle |z| = 1 and a log-power function Σ(z) at the points of Z such that Q(z) − Σ(z) is C t -smooth on the unit circle.
Definition 27. Let f (z) be analytic in the open unit disc. For ζ a point on the unit circle, we define the radial expansion of f at ζ with order t ∈ R as the smallest (in terms of numbers of monomials) log-power function σ(z) at ζ, provided it exists, such that when z = (1 − x)ζ and x tends to 0 + . The quantity σ(z) is written asymp(f (z), ζ, t). Now we are prepared to introduce the main theorem of the hybrid method.
Proposition 28 (Theorem 2 of [4]). Let f (z) be analytic in the open unit disc D, of finite global order a ≤ 0, and such that it admits a factorization f = P · Q, with P, Q analytic in D. Assume the following conditions on P and Q, relative to a finite set of points Z = {ζ 1 , ..., ζ m } on the unit circle ∂D: D1: The "Darboux factor" Q(z) is C s -smooth on ∂D (s ∈ Z ≥0 ). D2: The "singular factor" P (z) is analytically continuable to an indented domain of the form D = ∩ m j=1 (ζ j · ∆), where a ∆-domain is ∆(R, φ) := {z ∈ C | |z| < R, φ < arg(z − 1) < 2π − φ, z = 1} for some radius R > 1 and angle φ ∈ (0, π 2 ). For some non-negative real number t 0 , it admits, at any ζ j ∈ Z, an asymptotic form (a log-power expansion of class C t 0 ) where σ j (z) is a log-power function at 1. D3: t 0 > u 0 := ⌊ s+⌊a⌋ 2 ⌋. Then f admits radial expansions at every ζ j ∈ Z with order u 0 = ⌊ s+⌊a⌋ 2 ⌋. The coefficients of z n of f (z) satisfy: Now we turn to approximating the coefficients of W β (z), β > 1, to the order o(n −u 0 ) for some u 0 ∈ Z + which will be specified later as needed. We follow the hybrid method in close steps.
2.3.1. Darboux factor. . By the theorem we should choose a Darboux factor of C s -smooth for s = 2u 0 , note that the global order of W β (z) is zero. Provided the exp-log schema (8), we can factorize W β (z) into Since for l ≥ ⌊ 2u 0 +2 β ⌋, βl ≥ 2u 0 + 2 = s + 2, Li βl has all k = 0, · · · , s, its k-th derivatives admit a continuous extension onto the unit circle. Hence by Dirichlet's criterion as (9), V (z) is C s -smooth and we can take the Darboux factor as Q(z) = e V (z) .

2.3.2.
Singular factor. . Clearly we should take P (z) = e U (z) as the singular factor. U (z) = l<⌊ 2u 0 +2 β ⌋ h β,l 2 βl Li βl (z l ) as a truncation of the infinite sum, only has singularities at the l-th roots of unity for l ≤ ⌊ 2u 0 +2 β ⌋−1, by Lemma 22. This is to say P (z) is analytically continuable to the intersection of ∆domains pointed at those roots, which form the set Z. Also the lemma readily shows that P (z) admits the required asymptotic expansion to any order at each point of Z.
Hence by the theorem, W β (z) for any β > 1 admits a radial expansion at any point of Z with the chosen order u 0 and the hybrid method could give us the wanted asymptotics for W β,m once the radial expansions is calculated explicitly at each singularity. To simplify calculation, we set u 0 = ⌊β⌋ so that we only need to consider the expansion at l-th roots of unity for l ≤ 2u 0 +2 β − 1, which evaluates as follows In application, we mainly concern about the cases where β ∈ Z ≥2 and β → 2 − .

l=1
(1 − z) l l , to approximate W β (z) by logpower functions at z = 1 to the order u 0 = ⌊β⌋ is to approximate it to the order O(τ β ). Simply we have where H β,j = l≥1,βl−1 =j h β,l 2 βl ζ(βl − j)l j are convergent series. Hence in (13), we only need to care about the following terms We investigate the log-power expansion of these three terms separately.
First we write log τ as The other two terms B β (τ ) and δ β (τ ) do not involve log(1 − z), hence for large enough n, do not contribute to [z n ]W β (z) by the following lemma Lemma 29 (Lemma 1 of [4]). The general shape of coefficients of a logpower function is computable by the two rules: Note that Γ(z) has poles at negative integers which makes the first formula in the lemma coincide with the obvious fact that (1 − z) α , α ∈ Z ≥0 do not contribute to asymptotics of coefficients eventually. Combined with the above calculation, we get (14) the last of which recalls from l≥1 h β,l z l = log(I β (z)) = log j≥0 z j /(j!) β that h β,1 = 1 for any β ∈ R. In general, the coefficients h β,l can be computed by Faà di Bruno's formula. Hence we get the expansion for W β (z) at z = 1 in this shape.
Proof of Theorem 2. Now by the moment bound from Proposition 17, for P defined as (4) and f the random variable on {patitions of m} defined at the beginning of subsection 2.2, and for any c > 0, α > 0, we have . In particular since for any c > 0 there always exists α small enough such that (c − 1)α < 1/2, we have P (f < m c ) → 0, as m → ∞, which proves Theorem 2.

Proof of Theorem 3 by Wright's expression
Again by Markov's inequality, for any c > 0, 0 < β < 1 and expectation E P on the probability measure P defined in (4), 2 t log m and any 0 < t < 1 2 . Hence we need more precise asymptotics for W β,m , 0 < β < 1.
Let 1 2 < β < 1, we can split W β (z) as of (8) into where V β (z) = l≥2 h β,l 2 βl Li βl (z l ) and also note that h β,1 = 1. By calculation using the hybrid method in subsection 2.2, it is clear that For the first factor, by Lemma 22, we get (τ = − log z) Similar to 2.3.3, since δ β (τ ) (e δ β (τ ) ) only involves integer powers of (1−z), by Lemma 29 it does not contribute to the asymptotics of [z n ] exp 2 −β Li β (z) in order of n. Thus it is essential to approximate the coefficients of Together with the factorization (17) and asymptotics (18), this gives Now we focus on the asymptotics of [z n ]e 2 −β ζ(β) U β (z). We notice that functions of same type with U β were already handled in 1930s by E. M. Wright [12].