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Almost sure-sign convergence of Hardy-type Dirichlet series

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Abstract

Hartman proved in 1939 that the width of the largest possible strip in the complex plane on which a Dirichlet series \(\sum\nolimits_n {{a_n}{n^{ - s}}} \) is uniformly a.s.- sign convergent (i.e., \(\sum\nolimits_n {{\varepsilon _n}{a_n}{n^{ - s}}} \) converges uniformly for almost all sequences of signs εn = ±1) but does not convergent absolutely, equals 1/2. We study this result from a more modern point of view within the framework of so-called Hardytype Dirichlet series with values in a Banach space

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Authors and Affiliations

  1. Departamento de Matematica - Pab I, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, 1428, Buenos Aires, Argentina

    Daniel Carando

  2. IMAS - CONICET, Buenos Aires, Argentina

    Daniel Carando

  3. Institut für Mathematik, Universität Oldenburg, D-26111, Oldenburg, Germany

    Andreas Defant

  4. Instituto Universitario de Matem ática Pura y Aplicada, Universitat Politècnica de València, 46022, Valencia, Spain

    Pablo Sevilla-Peris

Authors
  1. Daniel Carando
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  2. Andreas Defant
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  3. Pablo Sevilla-Peris
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Correspondence to Daniel Carando.

Additional information

Supported by CONICET-PIP 11220130100329CO, PICT 2015–2299 and UBACyT 20020130100474BA.

Supported by MICINN MTM2017-83262-C2-1-P.

Supported by MICINN MTM2017-83262-C2-1-P and UPV-SP20120700.

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Carando, D., Defant, A. & Sevilla-Peris, P. Almost sure-sign convergence of Hardy-type Dirichlet series. JAMA 135, 225–247 (2018). https://doi.org/10.1007/s11854-018-0034-y

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  • DOI: https://doi.org/10.1007/s11854-018-0034-y

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