Numerical solution of some classical differential-difference equations

For differential-difference equations, we provide a method that gives numerical solutions accurate to hundreds or even thousands of digits. We illustrate with numerical solutions to three classical problems. With a few exceptions, previous claims of extended accuracy for these problems are found to be wrong.

[1]  K. Dickman On the frequency of numbers containing prime factors of a certain relative magnitude , 1930 .

[2]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[3]  V. Ramaswami On the number of positive integers less than $x$ and free of prime divisors greater than $x^c$ , 1949 .

[4]  de Ng Dick Bruijn On the number of uncancelled elements in the sieve of Eratosthenes , 1950 .

[5]  R. Bellman,et al.  On the Numerical Solution of a Differential-Difference Equation Arising in Analytic Number Theory: , 1962 .

[6]  Dura W. Sweeney On the computation of Euler’s constant , 1963 .

[7]  P Erdős,et al.  On the number of positive integers . . . , 1966 .

[8]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[9]  E. Wattel,et al.  On the numerical solution of a differential-difference equation arising in analytic number theory , 1969 .

[10]  Numerical solution of two differential-difference equations of analytic theory of numbers , 1969 .

[11]  Herbert Solomon,et al.  On random sequential packing in the plane and a conjecture of palasti , 1970, Journal of Applied Probability.

[12]  Karl K. Norton,et al.  Numbers with small prime factors : and the least kth power non-residue , 1971 .

[13]  P. Gillard,et al.  Evaluation of a Constant Associated with a Parking Problem , 1974 .

[14]  Miss A.O. Penney (b) , 1974, The New Yale Book of Quotations.

[15]  Donald E. Knuth,et al.  The Art of Computer Programming, Vol. 2 , 1981 .

[16]  H. Robbins,et al.  On the „Parking“ Problem , 1985 .

[17]  London,et al.  The distribution of quadratic and higher residues , .