On an Erdős–Kac-type conjecture of Elliott
Elliott and Halberstam proved that $\sum_{p \lt n} 2^{\omega(n-p)}$ is asymptotic to $\phi(n)$. In analogy to the Erdős–Kac theorem, Elliott conjectured that if one restricts the summation to primes p such that $\omega(n-p)\le 2 \log \log n+\lambda(2\log \log n)^{1/2}$ then the sum will be asymptoti...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Oxford University Press
2024
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| _version_ | 1817931488821248000 |
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| author | Gorodetsky, O Grimmelt, L |
| author_facet | Gorodetsky, O Grimmelt, L |
| author_sort | Gorodetsky, O |
| collection | OXFORD |
| description | Elliott and Halberstam proved that $\sum_{p \lt n} 2^{\omega(n-p)}$ is asymptotic to $\phi(n)$. In analogy to the Erdős–Kac theorem, Elliott conjectured that if one restricts the summation to primes p such that $\omega(n-p)\le 2 \log \log n+\lambda(2\log \log n)^{1/2}$ then the sum will be asymptotic to $\phi(n)\int_{-\infty}^{\lambda} \mathrm{e}^{-t^2/2}\,\mathrm{d}t/\sqrt{2\pi}$. We show that this conjecture follows from the Bombieri–Vinogradov theorem. We further prove a related result involving Poisson–Dirichlet distribution, employing deeper lying level of distribution results of the primes. |
| first_indexed | 2024-12-09T03:22:49Z |
| format | Journal article |
| id | oxford-uuid:b12c6ae9-e2c1-4b2d-8635-9b64bf0516e6 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:22:49Z |
| publishDate | 2024 |
| publisher | Oxford University Press |
| record_format | dspace |
| spelling | oxford-uuid:b12c6ae9-e2c1-4b2d-8635-9b64bf0516e62024-11-20T11:10:22ZOn an Erdős–Kac-type conjecture of ElliottJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b12c6ae9-e2c1-4b2d-8635-9b64bf0516e6EnglishSymplectic ElementsOxford University Press2024Gorodetsky, OGrimmelt, LElliott and Halberstam proved that $\sum_{p \lt n} 2^{\omega(n-p)}$ is asymptotic to $\phi(n)$. In analogy to the Erdős–Kac theorem, Elliott conjectured that if one restricts the summation to primes p such that $\omega(n-p)\le 2 \log \log n+\lambda(2\log \log n)^{1/2}$ then the sum will be asymptotic to $\phi(n)\int_{-\infty}^{\lambda} \mathrm{e}^{-t^2/2}\,\mathrm{d}t/\sqrt{2\pi}$. We show that this conjecture follows from the Bombieri–Vinogradov theorem. We further prove a related result involving Poisson–Dirichlet distribution, employing deeper lying level of distribution results of the primes. |
| spellingShingle | Gorodetsky, O Grimmelt, L On an Erdős–Kac-type conjecture of Elliott |
| title | On an Erdős–Kac-type conjecture of Elliott |
| title_full | On an Erdős–Kac-type conjecture of Elliott |
| title_fullStr | On an Erdős–Kac-type conjecture of Elliott |
| title_full_unstemmed | On an Erdős–Kac-type conjecture of Elliott |
| title_short | On an Erdős–Kac-type conjecture of Elliott |
| title_sort | on an erdos kac type conjecture of elliott |
| work_keys_str_mv | AT gorodetskyo onanerdoskactypeconjectureofelliott AT grimmeltl onanerdoskactypeconjectureofelliott |