Premium
On Bohr's theorem for general Dirichlet series
Author(s) -
Schoolmann I.
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800542
Subject(s) - mathematics , bohr model , general dirichlet series , dirichlet series , bounded function , holomorphic function , series (stratigraphy) , norm (philosophy) , dirichlet distribution , dirichlet eta function , pure mathematics , hardy space , complex plane , mathematical analysis , quantum mechanics , paleontology , physics , political science , law , boundary value problem , biology
We present quantitative versions of Bohr's theorem on general Dirichlet series D = ∑ a n e − λ n sassuming different assumptions on the frequency λ = ( λ n ) , including the conditions introduced by Bohr and Landau. Therefore, using the summation method by typical (first) means invented by M. Riesz, without any condition on λ, we give upper bounds for the norm of the partial sum operatorS N ( D ) : = ∑ n = 1 N a n ( D ) e − λ n sof length N on the spaceD ∞ e x t( λ )of all somewhere convergent λ‐Dirichlet series, which allow a holomorphic and bounded extension to the open right half plane [ R e > 0 ] . As a consequence for some classes of λ's we obtain a Montel theorem inD ∞ ( λ ) ; the space of all D ∈ D ∞ e x t( λ )which converge on [ R e > 0 ] . Moreover, following the ideas of Neder we give a construction of frequencies λ for whichD ∞ ( λ )fails to be complete.