On the Brun-Titchmarsh theorem
The Brun–Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C&am...
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| Format: | Journal article |
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Polskiej Akademii Nauk, Instytut Matematyczny
2013
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| _version_ | 1797103639419944960 |
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| author | Maynard, J |
| author_facet | Maynard, J |
| author_sort | Maynard, J |
| collection | OXFORD |
| description | The Brun–Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq≥8 and q is sufficiently large. Moreover, we also produce a lower bound of size x/(q1/2ϕ(q)) when logx/logq≥8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem. |
| first_indexed | 2024-03-07T06:22:56Z |
| format | Journal article |
| id | oxford-uuid:f34a2af4-c36d-4f9b-bad7-016a7a989203 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:22:56Z |
| publishDate | 2013 |
| publisher | Polskiej Akademii Nauk, Instytut Matematyczny |
| record_format | dspace |
| spelling | oxford-uuid:f34a2af4-c36d-4f9b-bad7-016a7a9892032022-03-27T12:10:55ZOn the Brun-Titchmarsh theoremJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f34a2af4-c36d-4f9b-bad7-016a7a989203Symplectic Elements at OxfordPolskiej Akademii Nauk, Instytut Matematyczny2013Maynard, JThe Brun–Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq≥8 and q is sufficiently large. Moreover, we also produce a lower bound of size x/(q1/2ϕ(q)) when logx/logq≥8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem. |
| spellingShingle | Maynard, J On the Brun-Titchmarsh theorem |
| title | On the Brun-Titchmarsh theorem |
| title_full | On the Brun-Titchmarsh theorem |
| title_fullStr | On the Brun-Titchmarsh theorem |
| title_full_unstemmed | On the Brun-Titchmarsh theorem |
| title_short | On the Brun-Titchmarsh theorem |
| title_sort | on the brun titchmarsh theorem |
| work_keys_str_mv | AT maynardj onthebruntitchmarshtheorem |