The Mathologer youtube channel is a treasure trove of mathematics made fun and accessible. His video on the Euler product formula is particularly clear and worth watching.
He even includes some really nice and easily derived consequences of Euler's product formula, including:
- there are infinitely many primes because the harmonic series diverges
- there are infinitely many primes because $\pi$ is not a rational number
- probability that two randomly chosen integers are coprime is $\frac{6}{\pi^2}$
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