Twin prime correlations from the pair correlation of Riemann zeros
We establish, via a heuristic Fourier inversion calculation, that the Hardy–Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height E on the critical line. Previously it was known that the Hardy–Littlewood conjecture...
| Hlavní autoři: | , |
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| Médium: | Journal article |
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IOP Publishing
2019
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| _version_ | 1826264760767741952 |
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| author | Keating, J Smith, D |
| author_facet | Keating, J Smith, D |
| author_sort | Keating, J |
| collection | OXFORD |
| description | We establish, via a heuristic Fourier inversion calculation, that the Hardy–Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height E on the critical line. Previously it was known that the Hardy–Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. An averaged form of the Hardy–Littlewood conjecture is obtained by inverting the limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula. |
| first_indexed | 2024-03-06T20:13:02Z |
| format | Journal article |
| id | oxford-uuid:2b3a3676-bbc1-487b-bbd6-e5c50f82aca2 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:13:02Z |
| publishDate | 2019 |
| publisher | IOP Publishing |
| record_format | dspace |
| spelling | oxford-uuid:2b3a3676-bbc1-487b-bbd6-e5c50f82aca22022-03-26T12:29:40ZTwin prime correlations from the pair correlation of Riemann zerosJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2b3a3676-bbc1-487b-bbd6-e5c50f82aca2Symplectic Elements at OxfordIOP Publishing2019Keating, JSmith, DWe establish, via a heuristic Fourier inversion calculation, that the Hardy–Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height E on the critical line. Previously it was known that the Hardy–Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. An averaged form of the Hardy–Littlewood conjecture is obtained by inverting the limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula. |
| spellingShingle | Keating, J Smith, D Twin prime correlations from the pair correlation of Riemann zeros |
| title | Twin prime correlations from the pair correlation of Riemann zeros |
| title_full | Twin prime correlations from the pair correlation of Riemann zeros |
| title_fullStr | Twin prime correlations from the pair correlation of Riemann zeros |
| title_full_unstemmed | Twin prime correlations from the pair correlation of Riemann zeros |
| title_short | Twin prime correlations from the pair correlation of Riemann zeros |
| title_sort | twin prime correlations from the pair correlation of riemann zeros |
| work_keys_str_mv | AT keatingj twinprimecorrelationsfromthepaircorrelationofriemannzeros AT smithd twinprimecorrelationsfromthepaircorrelationofriemannzeros |