Abstract
Sieving is all about extracting information from a sequence that is imbedded in a host sequence. The leading example is one of the primes: we start from the integers of the interval [1, x] and remove all the ones that are divisible by a prime not more than \(\sqrt{x}\). What remains are the primes from \((\sqrt{x},x]\) together with the integer 1.
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Notes
- 1.
The notations \(\Omega _p\) and \(\omega (p)\) are standard. They bear no relation with the functions \(\Omega \) and \(\omega \) that count the number of prime factors with and, respectively, without multiplicity.
- 2.
We take this opportunity to tell the readers that there are no known visual representation of Christian Goldbach, either drawing or painting, and that the ones that float on the web are all wrong (it is often Euler or H. Grassmann with specs, and sometimes it is B. Riemann).
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Ramaré, O. (2022). Montgomery’s Sieve. In: Excursions in Multiplicative Number Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-73169-4_29
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