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I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch theorem. Does anyone know where I could find such a reference? Much thanks in advance.

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See Theorem 4.1.11 and its proof in Niederreiter-Xing: Algebraic geometry in coding theory and cryptography.

Alternately, one can observe that the Riemann-Roch theorem in this setting is equivalent to the Poisson summation formula for the adeles of the function field (cf. Theorems 7-10 and 7-12 in Ramakrishnan-Valenza: Fourier analysis on number fields). The Poisson summation formula yields the functional equation plus location of poles of the zeta function, which in turn yield the claim via Cauchy's theorem on Taylor series of entire functions (cf. Theorem 4 in Section VII-6 of Weil: Basic number theory).

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