Exact asymptotics of positive solutions to Dickman equation

The paper considers the Dickman equation \begin{document}$\dot x (t)=-\frac{1}{t}\,x(t-1),$\end{document} for \begin{document} $t \to \infty $ \end{document} . The number theory uses what is called a Dickman (or Dickman -de Bruijn) function, which is the solution to this equation defined by an initial function \begin{document} $x(t)=1$ \end{document} if \begin{document} $0≤ t≤ 1$ \end{document} . The Dickman equation has two classes of asymptotically different positive solutions. The paper investigates their asymptotic behaviors in detail. A structure formula describing the asymptotic behavior of all solutions to the Dickman equation is given, an improvement of the well-known asymptotic behavior of the Dickman function, important in number theory, is derived and the problem of whether a given initial function defines dominant or subdominant solution is dealt with.

1. Introduction and preliminaries. The paper investigates the properties of solutions to the Dickman equatioṅ for t → ∞ where t ≥ t 0 > 0. Throughout the paper, the value t 0 may differ as different results are formulated and, in general, it is assumed to be sufficiently large in order to guarantee all the computations performed being well defined. This is mentioned in each particular case. A continuous function x : [t 0 − 1, ∞) → R is called a solution of (1) on [t 0 − 1, ∞) if it is continuously differentiable on [t 0 , ∞) and satisfies (1) for every t ∈ [t 0 , ∞) (at t = t 0 , the derivative is regarded as the derivative on the right). The initial problem x = ϕ(t), t ∈ [t 0 − 1, t 0 ), where ϕ is a continuous function, defines a unique solution x = x(t 0 , ϕ)(t), t ≥ t 0 − 1 of (1) such that A solution x of (1) on [t 0 − 1, ∞) is called positive if x(t) > 0 for every t ∈ [t 0 − 1, ∞), negative if x(t) < 0 for every t ∈ [t 0 − 1, ∞), and oscillating if it has arbitrarily large zeros on [t 0 − 1, ∞).
Let Ψ(y 1 , y 2 ) be the number of positive integers not exceeding y 1 having no prime divisors exceeding y 2 . Then, lim y→∞ Ψ(y t , y)y −t = ρ(t), t > 0 where ρ is what is called the Dickman function (or Dickman -de Bruin function because the latter author intensively studied it), defined for real t ≥ 0 by the relation As noted, e.g., in [20], the Dickman function was first studied by Dickman [12] and later by de Bruijn [6,7]. Differentiating (2), we can see that, assuming t 0 = 1, x = ρ(t) is a solution of equation (1) satisfying the unit initial condition on [0, 1]. Moreover, 0 < ρ(t) ≤ 1, |ρ (t)| ≤ 1, ρ(t) is nonincreasing, t ∈ [0, ∞), and ρ(t) ≤ 1/ t ! where · is the floor integer function. It is also known ( [6], see also [2]) that (throughout the paper, we use the well-known Landau order symbols O ("big" O) and o ("small" o) in computations with t → ∞). In [8] (see also [18]) an improved version of formula (3) is given: for all sufficiently large t and [24, p. 508] (formula for j (n+1) κ (u) where n = 0 and κ = 1) includes an improvement of formula (4) To the best of our knowledge, the formula (5) gives the best-possible asymptotic behavior of the function ρ published in available sources. For an overview of properties of the function ρ see also [18].
In the paper, we perform a qualitative analysis of the asymptotic behavior of the family of all solutions of (1) in terms of the theory of dominant and subdominant solutions to (1). We give the exact asymptotic behavior of the dominant solutions and sharp asymptotic behavior of the subdominant solutions. As a special case, we significantly improve the asymptotic behavior of the function ρ (being a subdominant solution using the below terminology) given by formula (4).
The paper is organized as follows. In part 1.1, the theory of dominant and subdominant solutions is shortly described. Then, the main results of the paper in part 2 are formulated where the existence of dominant and subdominant solutions to (1) and their asymptotic behaviors are discussed. Part 3 is devoted to important consequences of the main results. Namely, the structure formula describing the behavior of the family of all solutions to (1) is derived, the asymptotic behavior of Dickman function given by formula (5) is improved, and classes of initial functions defining either dominant or subdominant solutions are described. Proofs of the statements with the necessary auxiliary information are brought together in part 4.
1.1. Dominant and subdominant solutions. We will shortly overview the representation theory of solutions of equation (1) by what is called dominant and subdominant solutions. To this end, we adapt Theorems 8-10 and Definition 2 from [11] where more general equations than equation (1) are treated and formulate the relevant Theorems 1.1-1.3 and Definition 1.4. For this type of investigation, see also [15,16,17,19].
Moreover, every solution x = x(t) of (1) on [t 0 − 1, ∞) can be uniquely represented by the formula where the constant K depends on x.
Then, the formula (7) remains valid if x 1 is replaced byx 1 , the constant K is replaced by a constantK and x 2 is replaced byx 2 .   In what follows, we will work with what is called iterated logarithms. We define them as follows. The nth iterated logarithm ln n t (n ≥ 0) is defined as ln n t := ln(ln n−1 t), n ≥ 1, ln 0 t := t where we assume t > e n−1 for this definition to be correct.
In parts 2.1, 2.2 below, we use the terms "dominant" and "subdominant" solution to (1) in advance. When the existence of both types of solutions is proved, a verification of Definition 1.4 is simple and is given in part 2.3. (1). Let us look for a formal solution x = x(t) to (1) in the form of a power series with negative powers in a neighborhood of infinity

Dominant solutions to
with real coefficients C n defined by the following lemma.
Lemma 2.1. Coefficients C n of the formal solution x(t) = S(t) to (1) are defined by the formula where n ≥ 2 and the coefficient C 1 is chosen arbitrarily.
The proof of Lemma 2.1 is given in part 4.1.
To establish the convergence or divergence of the formal series S(t) defined by (8) is an open problem since the well-known criteria for the convergence or divergence of power series are not directly applicable. An attempt to utilize formula (9) to get estimates of the coefficients for the convergence/divergence tests to be applicable does not lead to applicable estimates.
The convergence/divergence problem explained in Remark 1 is the reason why we derive the following result on the existence of solutions to equation (1) asymptotically described by the formal series S(t) defined by (8).
Theorem 2.2. Let p ∈ N be fixed, let C 1 > 0 be fixed and let ε be a positive number such that ε > C p+1 . Then, there exists a dominant solution x(t) of (1) satisfying the inequality The proof of Theorem 2.2 is given in part 4.2. Let us remark that the asymptotic relation (10) is often written as (1). The statement of Theorem 2.2 implies the existence of positive solutions to (1) decreasing to zero as polynomials with negative powers do. In this part, we show that there exist positive solutions decreasing to zero even faster. Using the above terms, such solutions are called subdominant. We will describe them using a class of functions M β with specific asymptotic properties defined below.

Subdominant solutions to
The following lemma says that M β = ∅ by defining a class of functions satisfying all afore-mentioned properties. In addition, a sign property, necessary in the following investigation, is emphasized.
The proof of Lemma 2.4 is given in part 4.3.
The following result describes the asymptotic behavior of a subdominant solution.
Then, there exists a subdominant solution The proof of Theorem 2.5 is given in part 4.4.
3. Some consequences. The asymptotic behavior of both dominant and subdominant solutions to (1) together with representation formula (7) make it possible to formulate important properties of Dickman equation. Below, formula (7) is shown for the case in question, the asymptotic behavior of Dickman function is improved and a discussion on the sets of initial functions defining either dominant or subdominant solutions follows.
3.1. Structure formula describing the family of all solutions to (1). It is easy to write, utilizing formula (7) in Theorem 1.1, a structure formula describing the asymptotic behavior of all solutions to (1). As a solution x 1 we can take the solution described by formula (10) in Theorem 2.2, assuming that C 1 > 0 and ε > 0 are fixed, i.e., As a solution x 2 we can take the solution described in Theorem 2.5 and given by formula (20) in Remark 2, i.e.

3.2.
Improved asymptotic behavior of the Dickman function. The above results make an important contribution to the investigation of the Dickman equation by making it possible to improve the asymptotic behavior of Dickman function, given by formulas (3)- (5).
The following theorem provides the relevant statement and is proved in part 4.5.

On initial functions defining dominant and subdominant solutions.
Let x(t) = x(t 0 , ϕ)(t) be the unique solution of (1) with the initial data where ϕ : is equivalent to the following one where For t 0 = 1, the limit value is discussed, e.g., in [11,14,20,28]. If the limit exists and is finite, then Formula (31) is derived in [28] but without proving the existence of the limit (30). The existence of the limit can be deduced from the results in [11] and [14], or from formula (26), but these results cannot be used to derive formula (31). In [20], the authors gave an alternative proof of the limit equation (31) including the existence of the limit (30) (in connection with a discussion on the asymptotic convergence of solutions, we also refer to [3,4]). Let us also mention a recent paper [13], which describes a method for studying the asymptotic behavior of the dominant positive solutions to a similar class of scalar delay differential equations. Analyzing the initial-value problems (1), (28) and (29) with t 0 = 1, we conclude that the following theorem is valid. The exact asymptotic behavior of the dominant solution x(1, ϕ)(t) is specified in following theorem. Theorem 3.3. Let ϕ(t) > 0, t ∈ [0, 1] and C(1, ϕ) > 0. Then, the dominant solution x(1, ϕ)(t) of the initial-value problem (1), (28) for t → ∞ is asymptotically described as where the coefficients of the series in (32) are computed by formulas and x 2 (t) is arbitrary subdominant solution to (1).
The theorem is proved in part 4.6.

Remark 3.
Compare the asymptotic behavior of x(1, ϕ)(t) given by formulas (30) and (32). By (30) we get for t → ∞. Describing exactly the asymptotic behavior by a power series with negative powers, formula (32) substantially improves formula (34). The order of asymptotic accuracy is improved as well. A subdominant solution x 2 (t) can be described, e.g., by formula (25) and, therefore, lim t→∞ t n x 2 (t) = 0 for arbitrary n ∈ N.

4.
Proofs and additional material. This part contains proofs of the statements formulated above and the necessary auxiliary results and material. The proofs of the main results utilize the Ważewski method in a setting suitable to be applied to delayed differential equations [22]. This method is used in [9] to prove a theorem on the existence of solutions of delayed functional differential equations with graphs embedded in a previously defined domain. We employ the following particular case (but sufficient to determine the asymptotic behavior of dominant and subdominant solutions to (1)) of Theorem 1 in [9] adapted for equation (1).
For further computations, we need auxiliary formulas on asymptotic decompositions given in Lemma 4.1 and Lemma 4.2 in [10]. The following lemma summarizes the necessary formulas.
Lemma 4.2. Let reals σ and τ be fixed. Then, the following asymptotic representation holds for t → ∞. Let k ∈ N and reals σ, τ be fixed. Then, the following asymptotic representation holds for t → ∞.

4.1.
Proof of Lemma 2.1. In the below computations, for arbitrary non-negative integers n and λ, we define binomial number Matching the multipliers of the identical powers in (40) and (41), we obtain The last formula is equivalent to (9), i.e., a formal solution S(t) to (1) is of form (8) with coefficients defined by (9).

Proof of Theorem 2.2.
In the proof, we apply Theorem 4.1 with Verify inequality (35) first. Due to the assumptions of Theorem 4.1, we have ϕ(t + θ) < δ(t + θ) for every θ ∈ [−1, 0) and, therefore, So, it is sufficient to show that (43) holds. For the left-hand side of (43), we get and the right-hand side of (43) turns into (we use formula (9) with some technicalities being the same as in the proof of Lemma 2.1) Now it is easy to see that (43) will hold if p n=1 nC n t n+1 + This inequality is valid for all sufficiently large t and, therefore, inequality (43) holds. Now we show that inequality (36) holds as well. Since ϕ(t − 1) > π(t − 1), inequality (36) will be valid if Proceeding as above, we obtain inequality (44) where the symbol of inequality " > " is replaced by the opposite symbol " < " and ε replaced by −ε. Consequently, we conclude that inequality (36) holds if an inequality relevant to (45), i.e., is valid. Since the last inequality holds for all sufficiently large t, inequalities (46) and (36) hold as well.
Now it can be seen that for all t ∈ [t 0 , ∞) assuming t 0 is sufficiently large and (48) holds.
The above inequalities are equivalent with (19).