A differential delay equation arising from the sieve of Eratosthenes

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.