On the numerical solution of a differential-difference equation arising in analytic number theory
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- by J. van de Lune and E. Wattel PDF
- Math. Comp. 23 (1969), 417-421 Request permission
Abstract:
In the January 1962 issue of this Journal R. Bellman and B. Kotkin published a short paper under the same title as this one (cf. [1]). In that paper Bellman and Kotkin presented some of their results concerning the numerical computation of the continuous function $y(x)$, defined by \[ \begin {gathered} y(x) = 1(0 \leqq x \leqq 1) \hfill \\ y’ (x) = - \frac {1} {x} \cdot y(x - 1)(x > 1) \hfill \\ \end {gathered} \] Tables of $y(x)$ were given for $x = 1(0.0625)6$ and $x = 6(1)20$. In the process of extending these tables beyond $x = 20$ we discovered that the second table was rather inaccurate for all values of $x > 9$. Bellman and Kotkin found, for example, that $y(20) = 0.149 \cdot {10^{ - 8}}$, whereas the actual value of $y(20)$ can be shown to be smaller than ${10^{ - 20}}$. Moreover, in view of the method used by Bellman and Kotkin, one may expect that it would be quite time consuming to compute $y(x)$ for values of $x$ up to say $x = 1,000$. In this paper we describe a different method which enables us to compute $y(x)$ for values of $x$ up to about “as far as one would like."References
- 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 148248, DOI 10.1090/S0025-5718-1962-0148248-2
- 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 43838 For an extensive list of literature concerning the function $y(x)$ we refer to 3. J. van de Lune & E. Wattel, On the Frequency of Natural Numbers m whose Prime Divisors are all Smaller than ${m^\alpha }$, Mathematical Centre, Amsterdam, Report ZW 1968–007, 1968.
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Math. Comp. 23 (1969), 417-421
- MSC: Primary 65.70
- DOI: https://doi.org/10.1090/S0025-5718-1969-0247789-3
- MathSciNet review: 0247789