Fürstenberg’s proof on the infinity of primes revisited

Warning to teaching staff: summing infinite sequences of positive (>0) integers is difficult, but possible. For example is not a mistake and students must get full marks. Let me show you why.

A couple of months ago I was hoping to ‘cheat’ a proof of the Erdös discrepancy conjecture by using a variant of an idea of Fürstenberg’s proof on the infinitude of primes. Remember, Fürstenberg considered the integers with a new topology. Its open sets are for and .

This topology is metrizable. There are a couple of hand-waving arguments how this metric could look like. However, as far as I am aware of, there is so far no neat description in the literature. A couple of days ago, R. Lovas and I. Mezö have published a fairly straightforward proof that with induces the above topology.

Since the sequence converges to 0 in this topology. The partial sums of satisfy and thus .

R. Lovas and I. Mezö have collected more such observations in their note. What they did not mention explicitly, but what I consider interesting is that with the above metric, the integers become an ultrametric space. Without loss of generality we assume . Then are all divisors of m and n and thus they are divisors of m+n. Therefore . The strong triangle inequality now follows .

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