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Session 4. Banach Spaces and Operator Theory with Applications
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The Dirichlet-Bohr radius |
Domingo García, University of Valencia, Spain
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The talk is based on the joint work with D. Carando, A. Defant, M. Maestre and P. Sevilla-Peris
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Denote by \(\Omega(n)\) the number of prime divisors of \(n \in
\mathbb{N}\) (counted with multiplicities). For \(x\in \mathbb{N}\)
define the Dirichlet-Bohr radius \(L_x\) to be the best \(r>0\) such
that for every finite Dirichlet polynomial \(\sum_{n=1}^x a_n
n^{-s}\) we have \( \sum_{n=1}^x |a_n| r^{\Omega(n)} \leq \sup_{t\in
\mathbb{R}} \big|\sum_{n=1}^x a_n n^{-it}\big| \, . \) We prove
that the asymptotically correct order of \(L_x\) is \( (\log
x)^{1/4}x^{-1/8} \). Following Bohr's vision our proof links the
estimation of \(L_x\) with classical Bohr radii for holomorphic
functions in several variables. Moreover, we suggest a general
setting which allows to translate various results on Bohr radii in a
systematic way into results on Dirichlet Bohr radii, and vice versa.
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