On the Lindelöf hypothesis for general sequences
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916625070948352 |
|---|---|
| author | Broucke, Frederik Weishäupl, Sebastian |
| author_facet | Broucke, Frederik Weishäupl, Sebastian |
| contents | In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences which coincides with the usual Lindelöf hypothesis for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the ''generic'' admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of ''generic'': we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00239 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Lindelöf hypothesis for general sequences Broucke, Frederik Weishäupl, Sebastian Number Theory 11M26 (Primary) 11L03, 11N80 (Secondary) In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences which coincides with the usual Lindelöf hypothesis for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the ''generic'' admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of ''generic'': we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems. |
| title | On the Lindelöf hypothesis for general sequences |
| topic | Number Theory 11M26 (Primary) 11L03, 11N80 (Secondary) |
| url | https://arxiv.org/abs/2307.00239 |