Dickman Function Research Articles

Extreme values of derivatives of zeta and L-functions.

It is proved that as , uniformly for all positive integers , we have where. Here, is the Dickman function. We have and when , which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet L-functions. On the other hand, when assuming the Riemann hypothesis and the generalized Riemann hypothesis, we establish upper bounds for and . Furthermore, when assuming the Granville-Soundararajan conjecture is true, we establish the following asymptotic formulas: where is prime and isgiven.

Size biased sampling from the Dickman subordinator

The Dickman function and associated distribution play a prominent role in probabilistic number theory and in the theory of Poisson–Dirichlet distributions. These distributions in turn are closely related to properties of ordered jumps of subordinators. In the present paper we derive some remarkable invariance properties and other identities for the Dickman subordinator, related to the large jumps of the process. The implications of this for size-biased sampling from the Dickman distribution are drawn out.

A simple proof of the existence of the Dickman function

In this article we give a simple proof of the existence of the Dickman's function related with smooth numbers. We only use the concept of integral of a continuous function.

The Dickman subordinator, renewal theorems, and disordered systems

We consider the so-called Dickman subordinator, whose Levy measure has density $\frac{1} {x}$ restricted to the interval $(0,1)$. The marginal density of this process, known as the Dickman function, appears in many areas of mathematics, from number theory to combinatorics. In this paper, we study renewal processes in the domain of attraction of the Dickman subordinator, for which we prove local renewal theorems. We then present applications to marginally relevant disordered systems, such as pinning and directed polymer models, and prove sharp second moment estimates on their partition functions.

A note on an asymptotic expansion related to the Dickman function

In this paper we refine an asymptotic expansion given by Soundararajan related to the Dickman function. The result suggests a relatively simple approach to computing these integrals numerically.

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.

A family of multiple integrals connected with relatives of the Dickman function

In this paper we present an asymptotic expansion for a family of multiple integrals connected with relatives of the Dickman function. The coefficients of this expansion have a similar arithmetic structure as those appearing in Soundararajan's work on an analogous expansion for the Dickman function.

On the polynomial sharp upper estimate conjecture in 7-dimensional simplex

Because of its importance in number theory and singularity theory, the problem of finding a polynomial sharp upper estimate of the number of positive integral points in an n-dimensional (n≥3) polyhedron has received attention by a lot of mathematicians. The first named author proposed the Number Theoretic Conjecture for the upper estimate. The previous results on the Number Theoretic Conjecture in low dimension cases (n<7) are proved by using the sharp GLY conjecture which is true only for low dimensional case. Thus the proof cannot be generalized to high dimension. In this paper, we offer a uniform approach to prove the Number Theoretic Conjecture for all dimensions by simply using the induction method and the Yau–Zhang [19] estimates (see Lemmas 2.3–2.5). As a result, the Number Theoretic Conjecture is proven for n=7. An important estimate for all dimensions is also obtained (Propositions 3.1 and 3.2) which will be useful to prove the general case of the Number Theoretic Conjecture. As an application, we give a sharper estimate of the Dickman–De Bruijn function ψ(x,y) for 5≤y<19, compared with the result obtained by Ennola.

A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture (see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional (n ≥ 3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau Number Theoretic Conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5 ≤ y < 17, compared with the result obtained by Ennola.

A sharp polynomial estimate of positive integral points in a 4-dimensional tetrahedron and a sharp estimate of the Dickman-de Bruijn function

The estimate of integral points in right-angled simplices has many applications in number theory, complex geometry, toric variety and tropical geometry. In [24], [25], [27], the second author and other coworkers gave a sharp upper estimate that counts the number of positive integral points in n dimensional () real right-angled simplices with vertices whose distance to the origin are at least . A natural problem is how to form a new sharp estimate without the minimal distance assumption. In this paper, we formulate the Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. We have proved this conjecture for . This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of the Dickman-de Bruijn function for .

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function \psi(x,y) which is the number of positive integers \leq x and free of prime factors &gt;y . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( n\geq3 ) real right-angled simplices. In this paper, we prove this Number Theoretic Conjecture for n=5 . As an application, we give a sharp estimate of Dickman-De Bruijn function \psi(x,y) for 5\leq y&lt;13 .

Bounded size bias coupling: a Gamma function bound, and universal Dickman-function behavior

Under the assumption that the distribution of a nonnegative random variable $$X$$ admits a bounded coupling with its size biased version, we prove simple and strong concentration bounds. In particular the upper tail probability is shown to decay at least as fast as the reciprocal of a Gamma function, guaranteeing a moment generating function that converges everywhere. The class of infinitely divisible distributions with finite mean, whose Levy measure is supported on an interval contained in $$[0,c]$$ for some $$c < \infty $$ , forms a special case in which this upper bound is logarithmically sharp. In particular the asymptotic estimate for the Dickman function, that $$\rho (u) \approx u^{-u}$$ for large $$u$$ , is shown to be universal for this class. A special case of our bounds arises when $$X$$ is a sum of independent random variables, each admitting a 1-bounded size bias coupling. In this case, our bounds are comparable to Chernoff–Hoeffding bounds; however, ours are broader in scope, sharper for the upper tail, and equal for the lower tail. We discuss bounded and monotone couplings, give a sandwich principle, and show how this gives an easy conceptual proof that any finite positive mean sum of independent Bernoulli random variables admits a 1-bounded coupling with the same conditioned to be nonzero.

Nicolaas Govert de Bruijn, the enchanter of friable integers

N.G. de Bruijn carried out fundamental work on integers having only small prime factors and the Dickman–de Bruijn function that arises on computing the density of those integers. In this he used his earlier work on linear functionals and differential–difference equations. We review his relevant work and also some later improvements by others.

An asymptotic expansion related to the Dickman function

We prove a conjecture of Broadhurst on asymptotic expansions of certain polylogarithm type functions related to the Dickman function.

A Markov process associated with plot-size distribution in Czech Land Registry and its number-theoretic properties

The size distribution of land plots is a result of land allocation processes in the past. In the absence of regulation this is a Markov process leading an equilibrium described by a probabilistic equation used commonly in the insurance and financial mathematics. We support this claim by analyzing the distribution of two plot types, garden and build-up areas, in the Czech Land Registry pointing out the coincidence with the distribution of prime number factors described by Dickman function in the first case.

Asymptotics of Largest Components in Combinatorial Structures

Given integers m and n, we study the probability that structures of size n have all components of size at most m. The results are given in term of a generalized Dickman function of n/m.

On a Problem of Erdős Involving the Largest Prime Factor of n

Let P(n) denote the largest prime factor of an integer n (2), and let N (x) denote the number of natural numbers n such that 2 n x, and n does not divide P(n)!. We prove that where (u) is the Dickman-de Bruijn function. In terms of elementary functions we have thereby sharpening and correcting recent results of K. Ford and J.-M. De Koninck and N. Doyon.

Random minimal directed spanning trees and Dickman-type distributions

In Bhatt and Roy's minimal directed spanning tree construction fornrandom points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for largen) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

Random minimal directed spanning trees and Dickman-type distributions

In Bhatt and Roy's minimal directed spanning tree construction for n random points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large n) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

Quickselect and the Dickman Function

We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth-smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived.