The analogue of the Riemann hypothesis for curves over finite fields has been shown by André Weil (see also Roadmap to Riemann hypothesis for curves over finite fields) and further deep results (Weil conjectures) are known to hold for the zeta function in this case.

What I want to know is: what makes proving the Riemann hypothesis so much harder than its analogue for curves over finite fields? What makes it hard (probably impossible) to generalize the ideas that were used in the finite field case to a proof of the Riemann hypthesis?

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    $\begingroup$ let me guess: infiniteness? $\endgroup$ – Masacroso Jan 8 at 0:20
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    $\begingroup$ It should be possible to say on the $\mathbb{Z}$ case we need to approximate $\sum_{p^k \le x} \log(p) = \sum_{n \le x}\sum_{d | n} \mu(n/d)\log(d)$ on the $C/\mathbb{F}_q$ case it is $ \sum_{n \le x}\sum_{d | n} \mu(n/d)(q^d + O(q^{d/2}))$. The latter is obviously $\sum_{n \le x} (q^n+ O(q^{n(1/2+\epsilon)}))= \frac{q^{x+1}}{q-1}+O(q^{x(1/2+\epsilon)})$. So to go from Weil to RH you'd need to let $q \to 1$ and it doesn't behave well arithmetically and wrt asymptotics. $\endgroup$ – reuns Jan 8 at 1:40

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