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Zeta functions are typically analogues or generalizations of the Riemann zeta function. Examples include Dedekind zeta functions of number fields, and zeta functions of varieties over finite fields. They are typically initially defined as formal generating functions, but often admit analytic continuations.

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What is the spectral interpretation of the arithmetic zeta function?

First, it's worth mentioning that the above theorem refers to the (local) Hasse-Weil zeta function of a scheme $X$ of finite type over $\mathbb{F}_q$ rather than the arithmetic zeta function of a sche …
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