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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the numerical solution of a differential-difference equation arising in analytic number theory

Authors: J. van de Lune and E. Wattel
Journal: Math. Comp. 23 (1969), 417-421
MSC: Primary 65.70
MathSciNet review: 0247789
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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{displaymath}\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} \end{displaymath}

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."

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Article copyright: © Copyright 1969 American Mathematical Society