Linear constellations in primes with arithmetic restrictions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911248766992384 |
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| author | Frei, Christopher König, Joachim Tinková, Magdaléna |
| author_facet | Frei, Christopher König, Joachim Tinková, Magdaléna |
| contents | We prove analogues of the theorem of Green and Tao on linear constellations in primes, in which the primes under consideration are restricted by certain arithmetic conditions. Our first main result is conditional upon Hooley's Riemann hypothesis and imposes the extra condition that the primes have prescribed primitive roots. Our second main result is unconditional and imposes the extra condition that the primes have prescribed Artin symbols in given Galois number fields. In the appendix we present an application of the second result in inverse Galois theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_17441 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear constellations in primes with arithmetic restrictions Frei, Christopher König, Joachim Tinková, Magdaléna Number Theory 11P32 (11A07, 11R45, 11B30, 11R32) We prove analogues of the theorem of Green and Tao on linear constellations in primes, in which the primes under consideration are restricted by certain arithmetic conditions. Our first main result is conditional upon Hooley's Riemann hypothesis and imposes the extra condition that the primes have prescribed primitive roots. Our second main result is unconditional and imposes the extra condition that the primes have prescribed Artin symbols in given Galois number fields. In the appendix we present an application of the second result in inverse Galois theory. |
| title | Linear constellations in primes with arithmetic restrictions |
| topic | Number Theory 11P32 (11A07, 11R45, 11B30, 11R32) |
| url | https://arxiv.org/abs/2408.17441 |