Convergence and almost sure properties in Hardy spaces of Dirichlet series

Hardy–Orlicz Spaces of Dirichlet Series: An Interpolation Problem on Abscissae of Convergence

International Mathematics Research Notices ◽

10.1093/imrn/rnz242 ◽

2019 ◽

Author(s):

Maxime Bailleul ◽

Pascal Lefèvre ◽

Luis Rodríguez-Piazza

Keyword(s):

Dirichlet Series ◽

Hardy Spaces ◽

Upper Bound ◽

Interpolation Problem ◽

Orlicz Spaces ◽

Elementary Method ◽

Abscissa Of Convergence ◽

Upper Bound Estimate

Abstract The study of Hardy spaces of Dirichlet series denoted by $\mathscr{H}^p$ ($p\geq 1$) was initiated in [7] when $p=2$ and $p=\infty $, and in [2] for the general case. In this paper we introduce the Orlicz version of spaces of Dirichlet series $\mathscr{H}^\psi $. We focus on the case $\psi =\psi _q(t)=\exp (t^q)-1,$ and we compute the abscissa of convergence for these spaces. It turns out that its value is $\min \{1/q\,,1/2\}$ filling the gap between the case $\mathscr{H}^\infty $, where the abscissa is equal to $0$, and the case $\mathscr{H}^p$ for $p$ finite, where the abscissa is equal to $1/2$. The upper-bound estimate relies on an elementary method that applies to many spaces of Dirichlet series. This answers a question raised by Hedenmalm in [6].

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Hardy Spaces of Dirichlet Series

Texts and Readings in Mathematics - Diophantine Approximation and Dirichlet Series ◽

10.1007/978-981-15-9351-2_6 ◽

2020 ◽

pp. 145-196

Author(s):

Hervé Queffelec ◽

Martine Queffelec

Keyword(s):

Dirichlet Series ◽

Hardy Spaces

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Zeros of Functions in Bergman-Type Hilbert Spaces of Dirichlet Series

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-24745 ◽

2016 ◽

Vol 119 (2) ◽

pp. 237

Author(s):

Ole Fredrik Brevig

Keyword(s):

Hilbert Space ◽

Real Number ◽

Dirichlet Series ◽

Hilbert Spaces ◽

Hardy Spaces ◽

Weighted Spaces ◽

Number Of Divisors ◽

Zeros Of Functions

For a real number $\alpha$ the Hilbert space $\mathscr{D}_\alpha$ consists of those Dirichlet series $\sum_{n=1}^\infty a_n/n^s$ for which $\sum_{n=1}^\infty |a_n|^2/[d(n)]^\alpha < \infty$, where $d(n)$ denotes the number of divisors of $n$. We extend a theorem of Seip on the bounded zero sequences of functions in $\mathscr{D}_\alpha$ to the case $\alpha>0$. Generalizations to other weighted spaces of Dirichlet series are also discussed, as are partial results on the zeros of functions in the Hardy spaces of Dirichlet series $\mathscr{H}^p$, for $1\leq p <2$.

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