It reminds me of an idea I came up with when I was in middle school and was never quite able to rigorize. I was thinking about the product prod_(p prime) (1-1/p) and it occurred to me: (1-1/p) is the probability that a random natural number n is not divisible by p. Take the product over all primes: prod_p(1-1/p) is the probability that n is divisible by no primes. But of the infinitely many natural numbers, the only one that is divisible by no primes is 1! So this probability is 0.

When I grew up and learned some analysis, I found myself frustrated that I couldn’t make this proof rigorous, eg by using asymptotic density. (Although I did see hints of the idea in at least one proof I saw of some number theory fact.) At the same time I found myself a little bit miffed at zeta functions, which seemed messy and not very well motivated. So thank you for making clear to me how to make more sense of that intuition, and how this naturally leads to zeta functions.

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