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A note on Bohr’s theorem for Beurling integer systems

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Abstract
Given a sequence of frequencies {.n} n=1, a corresponding generalized Dirichlet series is of the form f (s) = n=1 ane-.n s. We are interested in multiplicatively generated systems, where each number e.n arises as a finite product of some given numbers {qn} n=1, 1 < qn. 8, referred to as Beurling primes. In the classical case, where.n = log n, Bohr's theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane {s >.}, then it actually converges uniformly in every half-plane {s >. + e}, e > 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 ofBeurling primes for which bothBohr'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.
Keywords
Bohr's theorem, Beurling integer systems

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Please use this url to cite or link to this publication:

MLA
Broucke, Frederik, et al. “A Note on Bohr’s Theorem for Beurling Integer Systems.” MATHEMATISCHE ANNALEN, 2024, doi:10.1007/s00208-023-02756-x.
APA
Broucke, F., Kouroupis, A., & Perfekt, K.-M. (2024). A note on Bohr’s theorem for Beurling integer systems. MATHEMATISCHE ANNALEN. https://doi.org/10.1007/s00208-023-02756-x
Chicago author-date
Broucke, Frederik, Athanasios Kouroupis, and Karl-Mikael Perfekt. 2024. “A Note on Bohr’s Theorem for Beurling Integer Systems.” MATHEMATISCHE ANNALEN. https://doi.org/10.1007/s00208-023-02756-x.
Chicago author-date (all authors)
Broucke, Frederik, Athanasios Kouroupis, and Karl-Mikael Perfekt. 2024. “A Note on Bohr’s Theorem for Beurling Integer Systems.” MATHEMATISCHE ANNALEN. doi:10.1007/s00208-023-02756-x.
Vancouver
1.
Broucke F, Kouroupis A, Perfekt K-M. A note on Bohr’s theorem for Beurling integer systems. MATHEMATISCHE ANNALEN. 2024;
IEEE
[1]
F. Broucke, A. Kouroupis, and K.-M. Perfekt, “A note on Bohr’s theorem for Beurling integer systems,” MATHEMATISCHE ANNALEN, 2024.
@article{01HF90YY5P51MFNG96PZNX9D7K,
  abstract     = {{Given a sequence of frequencies {.n} n=1, a corresponding generalized Dirichlet series is of the form f (s) = n=1 ane-.n s. We are interested in multiplicatively generated systems, where each number e.n arises as a finite product of some given numbers {qn} n=1, 1 < qn. 8, referred to as Beurling primes. In the classical case, where.n = log n, Bohr's theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane {s >.}, then it actually converges uniformly in every half-plane {s >. + e}, e > 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 ofBeurling primes for which bothBohr'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.}},
  author       = {{Broucke, Frederik and Kouroupis, Athanasios and Perfekt, Karl-Mikael}},
  issn         = {{0025-5831}},
  journal      = {{MATHEMATISCHE ANNALEN}},
  keywords     = {{Bohr's theorem,Beurling integer systems}},
  language     = {{eng}},
  pages        = {{15}},
  title        = {{A note on Bohr’s theorem for Beurling integer systems}},
  url          = {{http://doi.org/10.1007/s00208-023-02756-x}},
  year         = {{2024}},
}

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