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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 & 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.  Google Scholar

[2]

R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, 1963.  Google Scholar

[3]

H. Bereketoğlu and M. Pituk, Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003), 100-107.   Google Scholar

[4]

H. Bereketoğlu and F. Karakoç, Asymptotic constancy for impulsive delay differential equations, Dyn. Syst. Appl., 17 (2008), 71-83.   Google Scholar

[5]

D. Broadhurst, Dickman polylogarithms and their constants, arXiv: 1004. 0519v1 [math-ph] 4 Apr 2010, 1-11. Google Scholar

[6]

N. G. de Bruijn, On the number of positive integers $≤ x$ and free of prime factors $> y$, Indag. Math., 54 (1951), 50-60.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Diblík and M. Růžičková, Asymptotic behavior of solutions and positive solutions of differential delayed equations, Funct. Differ. Equ., 14 (2007), 83-105.   Google Scholar

[12]

K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys., 22A (1930), 1-14.   Google Scholar

[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.   Google Scholar

[14]

I. Györi and M. Pituk, Stability criteria for linear delay differential equations, J. Differential Equations, 10 (1997), 841-852.   Google Scholar

[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.   Google Scholar

[16]

E. Kozakiewicz, Über die nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Math. Nachr., 32 (1966), 107-113.  doi: 10.1002/mana.19660320112.  Google Scholar

[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.   Google Scholar

[18]

P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp.  Google Scholar

[19]

A. D. Myshkis, Linear Differential Equations with Retarded Arguments, Second edition. Izdat. "Nauka", Moscow, 1972.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

http://en.wikipedia.org/wiki/Dickman_function. Google Scholar

[26]

Nieuw Archief voor Wiskunde, Problem Section, 4/14 No 3 Nov. 1996, p. 429. Google Scholar

[27]

Nieuw Archief voor Wiskunde, Problem Section, 5/9 No 2 June 2008, p. 232. Google Scholar

[28]

Nieuw Archief voor Wiskunde, Problem Section, 5/11 No 1 March 2010, p. 76. Google Scholar

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.  Google Scholar

[2]

R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, 1963.  Google Scholar

[3]

H. Bereketoğlu and M. Pituk, Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays, Discrete Contin. Dyn. Syst., (2003), 100-107.   Google Scholar

[4]

H. Bereketoğlu and F. Karakoç, Asymptotic constancy for impulsive delay differential equations, Dyn. Syst. Appl., 17 (2008), 71-83.   Google Scholar

[5]

D. Broadhurst, Dickman polylogarithms and their constants, arXiv: 1004. 0519v1 [math-ph] 4 Apr 2010, 1-11. Google Scholar

[6]

N. G. de Bruijn, On the number of positive integers $≤ x$ and free of prime factors $> y$, Indag. Math., 54 (1951), 50-60.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Diblík and M. Růžičková, Asymptotic behavior of solutions and positive solutions of differential delayed equations, Funct. Differ. Equ., 14 (2007), 83-105.   Google Scholar

[12]

K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys., 22A (1930), 1-14.   Google Scholar

[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.   Google Scholar

[14]

I. Györi and M. Pituk, Stability criteria for linear delay differential equations, J. Differential Equations, 10 (1997), 841-852.   Google Scholar

[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.   Google Scholar

[16]

E. Kozakiewicz, Über die nichtschwingenden Lösungen einer linearen Differentialgleichung mit nacheilendem Argument, Math. Nachr., 32 (1966), 107-113.  doi: 10.1002/mana.19660320112.  Google Scholar

[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.   Google Scholar

[18]

P. Moree, Integers without large prime factors: From Ramanujan to de Bruijn, Integers, 14A (2014), Paper No. A5, 13 pp.  Google Scholar

[19]

A. D. Myshkis, Linear Differential Equations with Retarded Arguments, Second edition. Izdat. "Nauka", Moscow, 1972.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

http://en.wikipedia.org/wiki/Dickman_function. Google Scholar

[26]

Nieuw Archief voor Wiskunde, Problem Section, 4/14 No 3 Nov. 1996, p. 429. Google Scholar

[27]

Nieuw Archief voor Wiskunde, Problem Section, 5/9 No 2 June 2008, p. 232. Google Scholar

[28]

Nieuw Archief voor Wiskunde, Problem Section, 5/11 No 1 March 2010, p. 76. Google Scholar

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