Advertisement

Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 225–247 | Cite as

Almost sure-sign convergence of Hardy-type Dirichlet series

  • Daniel Carando
  • Andreas Defant
  • Pablo Sevilla-Peris
Article

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Aleman, J.-F. Olsen, and E. Saksman, Fourier multipliers for Hardy spaces of Dirichlet series, Int. Math. Res. Not. IMRN 16 (2014), 4368–4378.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    R. Balasubramanian, B. Calado, and H. Queffélec, The Bohr inequality for ordinary Dirichlet series Studia Math. 175 (2006), 285–304.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    F. Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136 (2002), 203–236.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    F. Bayart, A. Defant, L. Frerick, M. Maestre, and P. Sevilla-Peris, Monomial series expansion of Hp-functions and multipliers ofHp-Dirichlet series, Math. Ann. 368 (2017), 837–876.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda, The Bohr radius of the n-dimensional polydisk is equivalent to \(\sqrt {\left( {\log n} \right)/n} \), Adv. Math. 264 (2014), 726–746.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    F. Bayart, H. Queffélec, and K. Seip, Approximation numbers of composition operators on Hp spaces of Dirichlet series, Ann. Inst. Fourier (Grenoble) 66 (2016), 551–588.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    H. F. Bohnenblust and E. Hille. On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32 (1931), 600–622.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variablen in der Theorie der Dirichlet–schen Reihen \(\sum {\frac{{{a_n}}}{{{n^s}}}} \), Nachr. Ges.Wiss. Göttingen, Math. Phys. Kl., 1913, pp. 441–488.Google Scholar
  9. [9]
    D. Carando, A. Defant, and P. Sevilla-Peris, Bohr’s absolute convergence problem for Hp- Dirichlet series in Banach spaces, Anal. PDE 7 (2014), 513–527.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    D. Carando, A. Defant, and P. Sevilla-Peris, Some polynomial versions of cotype and applications, J. Funct. Anal. 270 (2016), 68–87.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    B. J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. (3) 53 (1986), 112–142.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    R. de la Bretèche. Sur l’ordre de grandeur des polynômes de Dirichlet, Acta Arith. 134 (2008), 141–148.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounäies, and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (2011), 485–497.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Defant, D. García, M. Maestre, and D. Pérez-García, Bohr’s strip for vector valued Dirichlet series, Math. Ann. 342 (2008), 533–555.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A. Defant, M. Maestre, and U. Schwarting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231 (2012), 2837–2857.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Defant and A. Pérez, Hardy spaces of vector-valued Dirichlet series, StudiaMath. (to appear), 2018 DOI: 10.4064/sm170303-26-7.Google Scholar
  17. [17]
    A. Defant, U. Schwarting, and P. Sevilla-Peris, Estimates for vector valued Dirichlet polynomials, Monatsh. Math. 175 (2014), 89–116.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995.CrossRefzbMATHGoogle Scholar
  19. [19]
    P. Hartman, On Dirichlet series involving random coefficients, Amer. J. Math. 61 (1939), 955–964.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L2(0, 1), Duke Math. J. 86 (1997), 1–37.CrossRefzbMATHGoogle Scholar
  21. [21]
    A. Hildebrand, and G. Tenenbaum, Integers without large prime factors, J. Thor. Nombres Bordeaux 5 (1993), 411–484.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    S. V. Konyagin and H. Queffélec, The translation 1/2 in the theory of Dirichlet series, Real Anal. Exchange 27 (2001/02) 155–175.MathSciNetzbMATHGoogle Scholar
  23. [23]
    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Springer-Verlag, Berlin, 1979.CrossRefzbMATHGoogle Scholar
  24. [24]
    B. Maurizi and H. Queffélec, Some remarks on the algebra of bounded Dirichlet series, J. Fourier Anal. Appl. 16 (2010), 676–692.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    H. Queffélec, H. Bohr’s vision of ordinary Dirichlet series; old and new results, J. Anal. 3 (1995), 43–60.MathSciNetzbMATHGoogle Scholar
  26. [26]
    H. Queffélec and M. Queffélec, Diophantine Approximation and Dirichlet Series, Hindustan Book Agency, New Delhi, 2013.CrossRefzbMATHGoogle Scholar
  27. [27]
    G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995.zbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  • Daniel Carando
    • 1
    • 2
  • Andreas Defant
    • 3
  • Pablo Sevilla-Peris
    • 4
  1. 1.Departamento de Matematica - Pab I, Facultad de Cs. Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.IMAS - CONICETBuenos AiresArgentina
  3. 3.Institut für MathematikUniversität OldenburgOldenburgGermany
  4. 4.Instituto Universitario de Matem ática Pura y AplicadaUniversitat Politècnica de ValènciaValenciaSpain

Personalised recommendations