Exact asymptotics of positive solutions to Dickman equation

Josef DiblÍk; Rigoberto Medina

doi:10.3934/dcdsb.2018007

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2018, Volume 23, Issue 1: 101-121. Doi: 10.3934/dcdsb.2018007

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Exact asymptotics of positive solutions to Dickman equation

Josef DiblÍk^1, , and
Rigoberto Medina^2,

1.
Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Veveří 331/95,602 00 Brno, Czech Republic
2.
Universidad de Los Lagos, Departamento de Ciencias Exactas, Casilla 933, Osorno, Chile

* Corresponding author:: J. DiblÍk
* Corresponding author:: J. DiblÍk

Received: July 2016

Published: November 2017

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Abstract

The paper considers the Dickman equation

$\dot x (t)=-\frac{1}{t}\,x(t-1),$

for $t \to \infty $ . 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 $x(t)=1$ if $0≤ t≤ 1$ . 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.

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Mathematics Subject Classification: Primary: 34K06, 34K25; Secondary: 11N25, 11A51.

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References

	E. Bach and R. Peralta , Asymptotic semismoothness probabilities, Math. Comp., 65 (1996) , 1701-1715. doi: 10.1090/S0025-5718-96-00775-2.
	R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, 1963.
	H. Bereketoğlu and M. Pituk , Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003) , 100-107.
	H. Bereketoğlu and F. Karakoç , Asymptotic constancy for impulsive delay differential equations, Dyn. Syst. Appl., 17 (2008) , 71-83.
	D. Broadhurst, Dickman polylogarithms and their constants, arXiv: 1004. 0519v1 [math-ph] 4 Apr 2010, 1-11.
	N. G. de Bruijn , On the number of positive integers $≤ x$ and free of prime factors $> y$, Indag. Math., 54 (1951) , 50-60.
	N. G. de Bruijn , On the number of positive integers $≤ x$ and free of prime factors $> y$ . Ⅱ, Indag. Math., 28 (1966) , 239-247.
	N. G. de Bruijn , The asymptotic behaviour of a function occurring in the theory of primes, J. Indian Math. Soc. (N.S.), 15 (1951) , 25-32.
	J. Diblík , A criterion for existence of positive solutions of systems of retarded functional differential equations, Nonlinear Anal., 38 (1999) , 327-339. doi: 10.1016/S0362-546X(98)00199-0.
	J. Diblík and N. Koksch , Positive solutions of the equation $\dot{x}(t)=-c(t)x(t-τ )$ in the critical case, J. Math. Anal. Appl., 250 (2000) , 635-659. doi: 10.1006/jmaa.2000.7008.
	J. Diblík and M. Růžičková , Asymptotic behavior of solutions and positive solutions of differential delayed equations, Funct. Differ. Equ., 14 (2007) , 83-105.
	K. Dickman , On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys., 22A (1930) , 1-14.
	I. Györi and M. Pituk , Asymptotic formulas for a scalar linear delay differential equation, Electron. J. Qual. Theory Differ. Equ., 2016 (2016) , 1-14.
	I. Györi and M. Pituk , Stability criteria for linear delay differential equations, J. Differential Equations, 10 (1997) , 841-852.
	E. Kozakiewicz , Über das asymptotische Verhalten der nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Wiss. Z. Humboldt Univ. Berlin, Math. Nat. R., 13 (1964) , 577-589.
	E. Kozakiewicz , Über die nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Math. Nachr., 32 (1966) , 107-113. doi: 10.1002/mana.19660320112.
	E. Kozakiewicz , Zur Abschätzung des Abklingens der nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Wiss. Z. Humboldt Univ. Berlin, Math. Nat. R., 15 (1966) , 675-676.
	P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp.
	A. D. Myshkis, Linear Differential Equations with Retarded Arguments, Second edition. Izdat. "Nauka", Moscow, 1972.
	M. Pituk and G. Röst , Large time behavior of a linear delay differential equation with asymptotically small coefficient, Bound. Value Probl., 2014 (2014) , 1-9. doi: 10.1186/1687-2770-2014-114.
	V. Ramaswami , On the number of positive integers less than $x$ and free of prime divisors greated than $x^c$, Bull. Amer. Math. Soc., 55 (1949) , 1122-1127. doi: 10.1090/S0002-9904-1949-09337-0.
	K. P. Rybakowski , Wa_zewski's principle for retarded functional differential equations, J. Differential Equations, 36 (1980) , 117-138. doi: 10.1016/0022-0396(80)90080-7.
	K. Soundararajan , An asymptotic expansion related to the Dickman function, The Ramanujan Journal, 29 (2012) , 25-30, arXiv:1005.3494v1. doi: 10.1007/s11139-011-9304-3.
	F. I. Wheeler , Two differential-difference equations arising in number theory, Trans. Amer. Math. Soc., 318 (1990) , 491-523. doi: 10.1090/S0002-9947-1990-0963247-X.
	http://en.wikipedia.org/wiki/Dickman_function.
	Nieuw Archief voor Wiskunde, Problem Section, 4/14 No 3 Nov. 1996, p. 429.
	Nieuw Archief voor Wiskunde, Problem Section, 5/9 No 2 June 2008, p. 232.
	Nieuw Archief voor Wiskunde, Problem Section, 5/11 No 1 March 2010, p. 76.