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

Author:
Jean-Marie-François 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

**[1]**N. G. de Bruijn,*On the number of positive integers ≤𝑥 and free of prime factors >𝑦*, Nederl. Acad. Wetensch. Proc. Ser. A.**54**(1951), 50–60. MR**0046375****[2]**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. MR**0043838****[3]**Karl K. Norton,*Numbers with small prime factors, and the least 𝑘th power non-residue*, Memoirs of the American Mathematical Society, No. 106, American Mathematical Society, Providence, R.I., 1971. MR**0286739****[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 and B. Kotkin,*On the numerical solution of a differential-difference equation arising in analytic number theory*, Math. Comp.**16**(1962), 473–475. MR**0148248**, https://doi.org/10.1090/S0025-5718-1962-0148248-2**[6]**J. van de Lune and E. Wattel,*On the numerical solution of a differential-difference equation arising in analytic number theory*, Math. Comp.**23**(1969), 417–421. MR**0247789**, https://doi.org/10.1090/S0025-5718-1969-0247789-3**[7]**M. P. van Ouwerkerk-Dijkers and J. Nuis,*On the asymptotic behavior of the solution of a differential-difference equation arising in number theory*, Math. Centrum Amsterdam Afd. Toegepaste Wisk. Rep. TN**50**(1968), 9. MR**0255945****[8]**L. I. Pál and G. Németh,*A statistical theory of lattice damage in solids irradiated by high-energy particles*, Nuovo Cimento (10)**12**(1959), 293–309 (English, with Italian summary). MR**0108918****[9]**L. Lewin,*Dilogarithms and associated functions*, Foreword by J. C. P. Miller, Macdonald, London, 1958. MR**0105524****[10]**J. H. Ahlberg, E. N. Nilson, and J. L. Walsh,*The theory of splines and their applications*, Academic Press, New York-London, 1967. MR**0239327****[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]**Edward W. Ng, C. J. Devine, and R. F. Tooper,*Chebyshev polynomial expansion of Bose-Einstein functions of orders 1 to 10*, Math. Comp.**23**(1969), 639–643. MR**0247739**, https://doi.org/10.1090/S0025-5718-1969-0247739-X**[13]**K. S. Kölbig, "Algorithm 327: Dilogarithm,"*Comm. Assoc. Comput. Mach.*, v. 11, 1968, pp. 270-271.**[14]**V. Boffi and R. Scozzafava,*Sull’equazione funzionale lineare 𝑓′(𝑥)=-𝐴(𝑥)𝑓(𝑥-1)*, Rend. Mat. e Appl. (5)**25**(1966), 402–410 (Italian). MR**0218702****[15]**H. Davenport and P. Erdös,*The distribution of quadratic and higher residues*, Publ. Math. Debrecen**2**(1952), 252–265. MR**0055368**

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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