Boundary Properties of Modified Zeta Functions and Function Spaces of Dirichlet Series
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Ordinary Dirichlet series, of which the Riemann zeta function is the most important, play a prominent role in classical analysis and number theory, and in modern mathematics. It is well-known that the Riemann zeta function has a single pole at the point s=1. The present thesis investigates both the behaviour of various zeta functions near this point and the function spaces of ordinary Dirichlet series they can be said to generate. Chapter 1 gives a comprehensive overview of the thesis and offers brief surveys of related results. Chapter 2 introduces a new scale of function spaces of Dirichlet series and explains the local behaviour of the reproducing kernels and establishes local embeddings into classical function spaces. Other such spaces are also considered, of which the Dirichlet-Hardy spaces are the most important. Chapter 3 determines the spaces spanned by the real parts of the boundary functions and distributions in the different settings. Chapter 4 characterises the local interpolating sequences for the Hilbert spaces under consideration. In the non-Hilbert spaces only partial results are obtained. Chapter 5 deals with a family of zeta functions corresponding to subsets of the integers. A complete characterisation of their behaviour close to the points = 1 is given in terms of lower norm bounds of integral operators with the zeta functions as kernels. Chapter 6 considers the results of the previous chapter under the additional hypothesis of arithmetic structure. The characterisations become simpler and more can be said.