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A probabilistic approach to a differential-difference equation arising in analytic number theory


Author: Jean-Marie-Fran├žois Chamayou
Journal: Math. Comp. 27 (1973), 197-203
MSC: Primary 65C05; Secondary 10K10
DOI: https://doi.org/10.1090/S0025-5718-1973-0336952-X
MathSciNet review: 0336952
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Abstract | References | Similar Articles | Additional Information

Abstract: The differential-difference equation

\begin{displaymath}\begin{array}{*{20}{c}} {tv'(t) + v(t - 1) = 0,} \hfill & {t ... ...nstant}},} \hfill & {0 \leqq t \leqq 1,} \hfill \\ \end{array} \end{displaymath}

can be solved by the Monte-Carlo method, for the initial condition $ v(t) = {e^{ - \gamma }},0 \leqq t \leqq 1$, where the $ v(t)$ represent the probability density of a random variable:

$\displaystyle t = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\prod\limits_{j = 1}^i {{x_j},} } $

where the $ {x_j}$ are independent and uniformly distributed on (0, 1).

References [Enhancements On Off] (What's this?)

  • [1] N. G. De Bruijn,"On the number of positive integers $ \leqq x$ and free of prime factors $ > y$," Nederl. Akad. Wetensch. Proc. Ser. A, v. 54 = Indag. Math., v. 13, 1951, pp. 50-60. MR 13, 724. MR 0046375 (13:724e)
  • [2] N. G. De Bruijn, "The asymptotic behaviour of a function occurring in the theory of primes," J. Indian Math. Soc., v. 15, 1951, pp. 25-32. MR 13, 326. MR 0043838 (13:326f)
  • [3] K. K. Norton, "Numbers with small prime factors and least Kth power non residue," Mem. Amer. Math. Soc., No. 106, 1971. MR 0286739 (44:3948)
  • [4] K. Dickman, "On the frequency of numbers containing prime factors of a certain relative magnitude," Ark. Mat. Astr. Fys., v. 22, A, 1930, pp. 1-14.
  • [5] R. Bellman & B. Kotkin, "On the numerical solution of a differential-difference equation arising in analytic number theory," Math. Comp., v. 16, 1962, pp. 473-475. MR 26 #5756. MR 0148248 (26:5756)
  • [6] J. van de Lune & E. Wattel, "On the numerical solution of a differential-difference equation arising in analytic number theory," Math. Comp., v. 23, 1969, pp. 417-421. MR 40 #1050. MR 0247789 (40:1050)
  • [7] M. P. van Ouwerkerk-Dijkers & J. Nuis, "On the asymptotic behaviour of the solution of a differential-difference equation arising in number theory," Math. Centrum Amsterdam Afd. Toegepaste Wisk. Rep. TN, v. 50, 1968, 9 pp. MR 41 #605. MR 0255945 (41:605)
  • [8] L. I. Pál & G. Németh, "A statistical theory of lattice damage in solids irradiated by high-energy particles," Nuovo Cimento (10), v. 12, 1959, pp. 293-309. MR 21 #7630. MR 0108918 (21:7630)
  • [9] L. Lewin, Dilogarithms and Associated Functions, MacDonald, LONDON, 1958. MR 21 #4264. MR 0105524 (21:4264)
  • [10] J. H. Ahlberg, E. N. Nilson & J. L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York, 1967. MR 39 #684. MR 0239327 (39:684)
  • [11] Y. L. Luke, The Special Functions and Their Approximations. Vol. 2, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR 40 #2909.
  • [12] E. W. Ng, C. J. Devine & R. F. Tooper, "Chebyshev polynomial expansion of Bose-Einstein functions of orders 1 to 10," Math. Comp., v. 23, 1969, pp. 639-643. MR 40 # 1002a. MR 0247739 (40:1002a)
  • [13] K. S. Kölbig, "Algorithm 327: Dilogarithm," Comm. Assoc. Comput. Mach., v. 11, 1968, pp. 270-271.
  • [14] V. Boffi & R. Scozzafava, "Sull' equazione funzionale lineare $ f'(x) = - A(x)f(x - 1)$," Rend. Mat. e Appl. (5), v. 25, 1966, pp. 402-410. MR 36 #1786. MR 0218702 (36:1786)
  • [15] H. Davenport & P. Erdös, "The distribution of quadratic and higher residues," Publ. Math. Debrecen, v. 2, 1951-52, pp. 252-265. MR 0055368 (14:1063h)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0336952-X
Keywords: Differential-difference equation, Monte-Carlo method, stochastic processes, elementary prime number theory, explicit machine computations
Article copyright: © Copyright 1973 American Mathematical Society

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