Numerical solution of some classical differentialdifference equations
Authors:
George Marsaglia, Arif Zaman and John C. W. Marsaglia
Journal:
Math. Comp. 53 (1989), 191201
MSC:
Primary 65L05; Secondary 65Q05
MathSciNet review:
969490
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: For differentialdifference equations, we provide a method that gives numerical solutions accurate to hundreds or even thousands of digits. We illustrate with numerical solutions to three classical problems. With a few exceptions, previous claims of extended accuracy for these problems are found to be wrong.
 [1]
R.
Bellman and B.
Kotkin, On the numerical solution of a
differentialdifference equation arising in analytic number
theory, Math. Comp. 16 (1962), 473–475. MR 0148248
(26 #5756), http://dx.doi.org/10.1090/S00255718196201482482
 [2]
N.
G. De Bruijn, On the number of uncancelled elements in the sieve of
Eratosthenes, Nederl. Akad. Wetensch., Proc. 53
(1950), 803–812 = Indagationes Math. 12, 247–256 (1950). MR 0035785
(12,11d)
 [3]
A. Buchstab, "Asymptotic estimates of a general numbertheoretic function," Mat. Sb. (N.S.), v. 44, 1937, pp. 12391246. (In Russian)
 [4]
B.
Edwin Blaisdell and Herbert
Solomon, On random sequential packing in the plane and a conjecture
of Palasti., J. Appl. Probability 7 (1970),
667–698. MR 0282389
(43 #8101)
 [5]
H.
Davenport and P.
Erdös, The distribution of quadratic and higher residues,
Publ. Math. Debrecen 2 (1952), 252–265. MR 0055368
(14,1063h)
 [6]
K. Dickman, "On the frequency of numbers containing prime factors of a certain relative magnitude," Ark. Mat. Astronom. Fys., v. 22A, 10, 1930, pp. 114.
 [7]
A.
Dvoretzky and H.
Robbins, On the “parking” problem, Magyar Tud.
Akad. Mat. Kutató Int. Közl. 9 (1964),
209–225 (English, with Russian summary). MR 0173275
(30 #3488)
 [8]
I.
S. Gradshteyn and I.
M. Ryzhik, Table of integrals, series, and products, Fourth
edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin. Translated
from the Russian by Scripta Technica, Inc. Translation edited by Alan
Jeffrey, Academic Press, New York, 1965. MR 0197789
(33 #5952)
 [9]
Donald
E. Knuth, The art of computer programming. Vol. 2, 2nd ed.,
AddisonWesley Publishing Co., Reading, Mass., 1981. Seminumerical
algorithms; AddisonWesley Series in Computer Science and Information
Processing. MR
633878 (83i:68003)
 [10]
M.
Lal and P.
Gillard, Evaluation of a constant associated
with a parking problem, Math. Comp. 28 (1974), 561–564. MR 0341814
(49 #6560), http://dx.doi.org/10.1090/S00255718197403418149
 [11]
M. Lal & P. Gillard, Numerical Solution of Two DifferentialDifference Equations of Analytic Theory of Numbers, Lecture Notes in Math., vol. 109, SpringerVerlag, Berlin and New York, 1969, pp. 179187.
 [12]
J.
van de Lune and E.
Wattel, On the numerical solution of a
differentialdifference equation arising in analytic number
theory, Math. Comp. 23 (1969), 417–421. MR 0247789
(40 #1050), http://dx.doi.org/10.1090/S00255718196902477893
 [13]
David
Mannion, Random spacefilling in one dimension, Magyar Tud.
Akad. Mat. Kutató Int. Közl 9 (1964),
143–154 (English, with Russian summary). MR 0177435
(31 #1698)
 [14]
V.
Ramaswami, On the number of positive integers
less than 𝑥 and free of prime divisors greater than
𝑥^{𝑐}, Bull. Amer. Math.
Soc. 55 (1949),
1122–1127. MR 0031958
(11,233f), http://dx.doi.org/10.1090/S000299041949093370
 [15]
Alfréd
Rényi, On a onedimensional problem concerning random space
filling, Magyar Tud. Akad. Mat. Kutató Int. Közl.
3 (1958), no. no 1/2, 109–127 (Hungarian, with
Russian and English summaries). MR 0104284
(21 #3039)
 [16]
Karl
K. Norton, Numbers with small prime factors, and the least
𝑘th power nonresidue, Memoirs of the American Mathematical
Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR 0286739
(44 #3948)
 [17]
Sigmund
Selberg, The number of cancelled elements in the sieve of
Eratosthenes, Norsk Mat. Tidsskr. 26 (1944),
79–84 (Norwegian). MR 0018691
(8,317a)
 [18]
Dura
W. Sweeney, On the computation of Euler’s
constant, Math. Comp. 17 (1963), 170–178. MR 0160308
(28 #3522), http://dx.doi.org/10.1090/S0025571819630160308X
 [1]
 R. Bellman & B. Kotkin, "On the numerical solution of a differentialdifference equation arising in analytic number theory," Math. Comp., v. 16, 1962, pp. 473475. MR 0148248 (26:5756)
 [2]
 N. G. De Bruijn, "On the number of uncancelled elements in the sieve of Eratosthenes," Nederl. Akad. Wetensch. Proc., v. 53, 1950, pp. 803812; Indag. Math., v. 12, 1950, pp. 247256. MR 0035785 (12:11d)
 [3]
 A. Buchstab, "Asymptotic estimates of a general numbertheoretic function," Mat. Sb. (N.S.), v. 44, 1937, pp. 12391246. (In Russian)
 [4]
 B. E. Blaisdell & H. Solomon, "On random sequential packing in the plane and a conjecture of Palesti," J. Appl. Probab., v. 7, 1970, pp. 667698. MR 0282389 (43:8101)
 [5]
 H. Davenport & P. ErdöS, "The distribution of quadratic and higher residues," Publ. Math. Debrecen, v. 2, 19511952, pp. 252265. MR 0055368 (14:1063h)
 [6]
 K. Dickman, "On the frequency of numbers containing prime factors of a certain relative magnitude," Ark. Mat. Astronom. Fys., v. 22A, 10, 1930, pp. 114.
 [7]
 A. Dvoretsky & H. Robbins, "On the 'parking' problem," Publ. Math. Inst. Hungar. Acad. Sci., v. 9, 1964, pp. 209226. MR 0173275 (30:3488)
 [8]
 I. S. Gradshteyn & I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York and London, 1965. MR 0197789 (33:5952)
 [9]
 D. E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, AddisonWesley, Reading, Mass., 1981. MR 633878 (83i:68003)
 [10]
 M. Lal & P. Gillard, "Evaluation of a constant associated with a parking problem," Math. Comp., v. 28, 1974, pp. 561564. MR 0341814 (49:6560)
 [11]
 M. Lal & P. Gillard, Numerical Solution of Two DifferentialDifference Equations of Analytic Theory of Numbers, Lecture Notes in Math., vol. 109, SpringerVerlag, Berlin and New York, 1969, pp. 179187.
 [12]
 J. Van De Lune & E. Wattel, "On the numerical solution of a differentialdifference equation arising in analytic number theory," Math. Comp., v. 23, 1969, pp. 417421. MR 0247789 (40:1050)
 [13]
 David Manion, "Random spacefilling in one dimension," Publ. Math. Inst. Hungar. Acad. Sci., v. 9, 1964, pp. 143153. MR 0177435 (31:1698)
 [14]
 V. Ramaswami, "On the number of positive integers less than x and free of prime factors greater than ," Bull. Amer. Math. Soc., v. 55, 1949, pp. 11221127. MR 0031958 (11:233f)
 [15]
 Alfred Renyi, "On a onedimensional problem concerning random space filling," Publ. Math. Inst. Hungar. Acad. Sci., v. 3, 1958, pp. 109127. (In Hungarian) MR 0104284 (21:3039)
 [16]
 K. K. Norton, Numbers with Small Prime Factors and Least kth Power NonResidues, Mem. Amer. Math. Soc., vol. 106, Amer. Math. Soc., Providence, R.I., 1971, pp. 927. MR 0286739 (44:3948)
 [17]
 S. Selberg, "The number of cancelled elements in the sieve of Eratosthenes," Norsk. Mat. Tidsskr., v. 26, 1944, pp. 7984. (In Norwegian) MR 0018691 (8:317a)
 [18]
 D. W. Sweeney, "On the computation of Euler's constant," Math. Comp., v. 17, 1963, pp. 170178. MR 0160308 (28:3522)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65L05,
65Q05
Retrieve articles in all journals
with MSC:
65L05,
65Q05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909694903
PII:
S 00255718(1989)09694903
Keywords:
Difference equations,
differentialdifference equations,
numerical solutions
Article copyright:
© Copyright 1989 American Mathematical Society
