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Mean values, order, and convergence in the Hardy space of Dirichlet series
Abstract
Let ℋ2 denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if f in ℋ2 has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values |f|ℋ22 = limσ→ 0+ limT→∞ 1/2T ∫−TT lvert f(σ+it) rvert2 dt. We investigate to what extent the assumption that the analytic continuation be bounded can be relaxed, and obtain optimal results in two different directions. The first direction concerns the order of the analytic continuation, and yields as a corollary a version of the classical Landau--Schnee convergence theorem for ℋ2. The second direction concerns the abscissa of convergence, and resolves a problem posed by McCarthy.
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BibTeX
@misc{brevig2026mean,
title = {{Mean values, order, and convergence in the Hardy space of Dirichlet series}},
author = {Ole Fredrik Brevig and Athanasios Kouroupis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35496},
eprint = {2609.35496},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35496},
}
RIS
TY - UNPB
TI - Mean values, order, and convergence in the Hardy space of Dirichlet series
AU - Ole Fredrik Brevig
AU - Athanasios Kouroupis
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35496
UR - https://arxiv.org/abs/2609.35496
AB - Let ℋ^2 denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if f in ℋ^2 has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values |f|_ℋ^2^2 = lim_σ→ 0^+ lim_T→∞ 1/2T ∫_−T^T lvert f(σ+it) rvert^2 dt. We investigate to what extent the assumption that the analytic continuation be bounded can be relaxed, and obtain optimal results in two different directions. The first direction concerns the order of the analytic continuation, and yields as a corollary a version of the classical Landau--Schnee convergence theorem for ℋ^2. The second direction concerns the abscissa of convergence, and resolves a problem posed by McCarthy.
ER -