A differential delay equation arising from the sieve of Eratosthenes

Cheer, A. Y.; Goldston, D. A.

doi:10.1090/S0025-5718-1990-1023043-8

A differential delay equation arising from the sieve of Eratosthenes
HTML articles powered by AMS MathViewer

by A. Y. Cheer and D. A. Goldston PDF

Math. Comp. 55 (1990), 129-141 Request permission

Abstract:

The differential delay equation defined by $\omega (u) = 1/u$ for $1 \leq u \leq 2$ and $(u\omega (u))’ = \omega (u - 1)$ for $u \geq 2$ was introduced by Buchstab in connection with an asymptotic formula for the number of uncanceled terms in the sieve of Eratosthenes. Maier has recently used this result to show there is unexpected irregularity in the distribution of primes in short intervals. The function $\omega (u)$ is studied in this paper using numerical and analytical techniques. The results are applied to give some numerical constants in Maier’s theorem.

References

Asymptotic estimates of a general number-theoretic function

44

N. G. De Bruijn, On the number of uncancelled elements in the sieve of Eratosthenes, Nederl. Akad. Wetensch., Proc. 53 (1950), 803–812 = Indagationes Math. 12, 247–256 (1950). MR 35785
F. Grupp and H.-E. Richert, The functions of the linear sieve, J. Number Theory 22 (1986), no. 2, 208–239. MR 826952, DOI 10.1016/0022-314X(86)90070-3
Heini Halberstam and Klaus Friedrich Roth, Sequences, 2nd ed., Springer-Verlag, New York-Berlin, 1983. MR 687978

Estimation of an integral

4

Helmut Maier, Primes in short intervals, Michigan Math. J. 32 (1985), no. 2, 221–225. MR 783576, DOI 10.1307/mmj/1029003189