Exact asymptotics of positive solutions to Dickman equation

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January 2018, 23(1): 101-121. doi: 10.3934/dcdsb.2018007

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

Received July 2016 Published January 2018

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.

Keywords: Dickman equation, positive solution, dominant solution, subdominant solution, large time behavior, asymptotic representation, delayed differential equation.

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

Citation: Josef DiblÍk, Rigoberto Medina. Exact asymptotics of positive solutions to Dickman equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 101-121. doi: 10.3934/dcdsb.2018007

References:

[1]	E. Bach and R. Peralta, Asymptotic semismoothness probabilities, Math. Comp., 65 (1996), 1701-1715. doi: 10.1090/S0025-5718-96-00775-2.
[2]	R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, 1963.
[3]	H. Bereketoğlu and M. Pituk, Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003), 100-107.
[4]	H. Bereketoğlu and F. Karakoç, Asymptotic constancy for impulsive delay differential equations, Dyn. Syst. Appl., 17 (2008), 71-83.
[5]	D. Broadhurst, Dickman polylogarithms and their constants, arXiv: 1004. 0519v1 [math-ph] 4 Apr 2010, 1-11.
[6]	N. G. de Bruijn, On the number of positive integers $≤ x$ and free of prime factors $> y$, Indag. Math., 54 (1951), 50-60.
[7]	N. G. de Bruijn, On the number of positive integers $≤ x$ and free of prime factors $> y$. Ⅱ, Indag. Math., 28 (1966), 239-247.
[8]	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.
[9]	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.
[10]	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.
[11]	J. Diblík and M. Růžičková, Asymptotic behavior of solutions and positive solutions of differential delayed equations, Funct. Differ. Equ., 14 (2007), 83-105.
[12]	K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys., 22A (1930), 1-14.
[13]	I. Györi and M. Pituk, Asymptotic formulas for a scalar linear delay differential equation, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1-14.
[14]	I. Györi and M. Pituk, Stability criteria for linear delay differential equations, J. Differential Equations, 10 (1997), 841-852.
[15]	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.
[16]	E. Kozakiewicz, Über die nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Math. Nachr., 32 (1966), 107-113. doi: 10.1002/mana.19660320112.
[17]	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.
[18]	P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp.
[19]	A. D. Myshkis, Linear Differential Equations with Retarded Arguments, Second edition. Izdat. "Nauka", Moscow, 1972.
[20]	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.
[21]	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.
[22]	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.
[23]	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.
[24]	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.
[25]	http://en.wikipedia.org/wiki/Dickman_function.
[26]	Nieuw Archief voor Wiskunde, Problem Section, 4/14 No 3 Nov. 1996, p. 429.
[27]	Nieuw Archief voor Wiskunde, Problem Section, 5/9 No 2 June 2008, p. 232.
[28]	Nieuw Archief voor Wiskunde, Problem Section, 5/11 No 1 March 2010, p. 76.

show all references

References:

[1]	E. Bach and R. Peralta, Asymptotic semismoothness probabilities, Math. Comp., 65 (1996), 1701-1715. doi: 10.1090/S0025-5718-96-00775-2.
[2]	R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, 1963.
[3]	H. Bereketoğlu and M. Pituk, Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003), 100-107.
[4]	H. Bereketoğlu and F. Karakoç, Asymptotic constancy for impulsive delay differential equations, Dyn. Syst. Appl., 17 (2008), 71-83.
[5]	D. Broadhurst, Dickman polylogarithms and their constants, arXiv: 1004. 0519v1 [math-ph] 4 Apr 2010, 1-11.
[6]	N. G. de Bruijn, On the number of positive integers $≤ x$ and free of prime factors $> y$, Indag. Math., 54 (1951), 50-60.
[7]	N. G. de Bruijn, On the number of positive integers $≤ x$ and free of prime factors $> y$. Ⅱ, Indag. Math., 28 (1966), 239-247.
[8]	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.
[9]	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.
[10]	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.
[11]	J. Diblík and M. Růžičková, Asymptotic behavior of solutions and positive solutions of differential delayed equations, Funct. Differ. Equ., 14 (2007), 83-105.
[12]	K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys., 22A (1930), 1-14.
[13]	I. Györi and M. Pituk, Asymptotic formulas for a scalar linear delay differential equation, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 1-14.
[14]	I. Györi and M. Pituk, Stability criteria for linear delay differential equations, J. Differential Equations, 10 (1997), 841-852.
[15]	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.
[16]	E. Kozakiewicz, Über die nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Math. Nachr., 32 (1966), 107-113. doi: 10.1002/mana.19660320112.
[17]	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.
[18]	P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp.
[19]	A. D. Myshkis, Linear Differential Equations with Retarded Arguments, Second edition. Izdat. "Nauka", Moscow, 1972.
[20]	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.
[21]	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.
[22]	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.
[23]	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.
[24]	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.
[25]	http://en.wikipedia.org/wiki/Dickman_function.
[26]	Nieuw Archief voor Wiskunde, Problem Section, 4/14 No 3 Nov. 1996, p. 429.
[27]	Nieuw Archief voor Wiskunde, Problem Section, 5/9 No 2 June 2008, p. 232.
[28]	Nieuw Archief voor Wiskunde, Problem Section, 5/11 No 1 March 2010, p. 76.

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