On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory:
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Summary. The computational solution of a certain class of differential-difference equations requires numerical procedures involving an extremely high degree of precision to obtain accurate results over a large range of the independent variable. One method of solution uses an iterative procedure which relates the differentialdifference equation over a large range to a system of ordinary differential equations over a limited range. When the characteristic roots of the related system indicate borderline stability, it is evident that small perturbations in obtaining successive initial values eventually grow out of control as the system increases. To investigate this phenomenon, we examine the equation u' (x) =-u (x -1) /x. arising in analytic number theory.
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