A note on Bohr's theorem for Beurling integer systems
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914787612426240 |
|---|---|
| author | Broucke, Frederik Kouroupis, Athanasios Perfekt, Karl-Mikael |
| author_facet | Broucke, Frederik Kouroupis, Athanasios Perfekt, Karl-Mikael |
| contents | Given a sequence of frequencies $\{λ_n\}_{n\geq1}$, a corresponding generalized Dirichlet series is of the form $f(s)=\sum_{n\geq 1}a_ne^{-λ_ns}$. We are interested in multiplicatively generated systems, where each number $e^{λ_n}$ arises as a finite product of some given numbers $\{q_n\}_{n\geq 1}$, $1 < q_n \to \infty$, referred to as Beurling primes. In the classical case, where $λ_n = \log n$, Bohr's theorem holds: if $f$ converges somewhere and has an analytic extension which is bounded in a half-plane $\{\Re s> θ\}$, then it actually converges uniformly in every half-plane $\{\Re s> θ+\varepsilon\}$, $\varepsilon>0$. We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr's condition, and therefore the theorem. Applying our technique in conjunction with a probabilistic method, we find a system of Beurling primes for which both Bohr's theorem and the Riemann hypothesis are valid. This provides a counterexample to a conjecture of H. Helson concerning outer functions in Hardy spaces of generalized Dirichlet series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11782 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on Bohr's theorem for Beurling integer systems Broucke, Frederik Kouroupis, Athanasios Perfekt, Karl-Mikael Number Theory Complex Variables Functional Analysis Given a sequence of frequencies $\{λ_n\}_{n\geq1}$, a corresponding generalized Dirichlet series is of the form $f(s)=\sum_{n\geq 1}a_ne^{-λ_ns}$. We are interested in multiplicatively generated systems, where each number $e^{λ_n}$ arises as a finite product of some given numbers $\{q_n\}_{n\geq 1}$, $1 < q_n \to \infty$, referred to as Beurling primes. In the classical case, where $λ_n = \log n$, Bohr's theorem holds: if $f$ converges somewhere and has an analytic extension which is bounded in a half-plane $\{\Re s> θ\}$, then it actually converges uniformly in every half-plane $\{\Re s> θ+\varepsilon\}$, $\varepsilon>0$. We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr's condition, and therefore the theorem. Applying our technique in conjunction with a probabilistic method, we find a system of Beurling primes for which both Bohr's theorem and the Riemann hypothesis are valid. This provides a counterexample to a conjecture of H. Helson concerning outer functions in Hardy spaces of generalized Dirichlet series. |
| title | A note on Bohr's theorem for Beurling integer systems |
| topic | Number Theory Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2301.11782 |