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

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."