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A differential delay equation arising from the sieve of Eratosthenes


Authors: A. Y. Cheer and D. A. Goldston
Journal: Math. Comp. 55 (1990), 129-141
MSC: Primary 11N35; Secondary 34K05
DOI: https://doi.org/10.1090/S0025-5718-1990-1023043-8
MathSciNet review: 1023043
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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 [Enhancements On Off] (What's this?)

  • [1] A. A. Buchstab, Asymptotic estimates of a general number-theoretic function, Mat. Sb. 44 (1937), 1239-1246. (Russian; German summary)
  • [2] N. G. de Bruijn, On the number of uncancelled elements in the sieve of Eratosthenes, Nederl. Akad. Wetensch. Proc. 53 (1950), 803-812. MR 0035785 (12:11d)
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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-1023043-8
Article copyright: © Copyright 1990 American Mathematical Society

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