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from Part 1 - Bohr’s Problem and Complex Analysis on Polydiscs
Published online by Cambridge University Press: 19 July 2019
We establish a bijection between Dirichlet series and formal power series through Bohr’s transform. This is one of the main tools all along the text and relies on the fact that by the fundamental theorem of arithmetic every natural number has a unique decomposition as a product of prime numbers. In this way, to each such number a multi-index can be assigned (and vice-versa). With this we show that the space of bounded holomorphic functions on B_{c0} and \mathcal{H}_\infty are isomorphic as Banach spaces. This means that to every holomorphic function corresponds a Dirichlet series in such a way that the monomial and the Dirichlet coefficients are identified. We consider m-homogenous Dirichlet series: those having non-zero coefficients only if n has exactly m prime divisors (counted with multiplicity) and show that the space of such Dirichlet series is isometrically isomorphic to the space of m-homogeneous polynomials on c0.
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