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Euler product

infinitude of primes
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Mathematics Subject Classification

11A05 no label found11A51 no label found


You mention how at $s=2$, the fact that the product of all primes yields an irrational number is a proof of the infinitude of primes.

That's true, but there's an easier way to get this result from the Euler product, which doesn't depend on comparatively advanced results like the irrationality of $\Pi$:

The $s=1$ case corresponds to the harmonic series. Proving that this diverges is a standard exercise which students will be familiar with. If there were only finitely many primes, the Euler product would be a nonzero rational number and so the harmonic series would have to converge.

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