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 | References | Similar Articles | Additional Information
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.
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A. A. Buchstab, Asymptotic estimates of a general number-theoretic function, Mat. Sb. 44 (1937), 1239-1246. (Russian; German summary)
- 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 https://doi.org/10.1016/0022-314X%2886%2990070-3
- Heini Halberstam and Klaus Friedrich Roth, Sequences, 2nd ed., Springer-Verlag, New York-Berlin, 1983. MR 687978 Loo-Keng Hua, Estimation of an integral, Sci. Sinica 4 (1951), 393-402.
- Helmut Maier, Primes in short intervals, Michigan Math. J. 32 (1985), no. 2, 221–225. MR 783576, DOI https://doi.org/10.1307/mmj/1029003189
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Article copyright:
© Copyright 1990
American Mathematical Society