Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces
Oliver, Thomas David (2014) Higher dimensional adeles, mean-periodicity and zeta functions of arithmetic surfaces. PhD thesis, University of Nottingham.
This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodicity correspondence”, which provides a passage between nicely behaved arithmetic schemes and mean-periodic functions in certain functional spaces. In what follows, there are two major themes.
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