Someone asked on Twitter:

Indeed this series diverges, just as the tweeter says: there’s a positive-density subset of n such that the summand exceeds .

More subtle: what about

This should still diverge. Argument: the probability that a real number x chosen uniformly from a large interval has is on order , not ; so there will be a subset of integers with density on order where the summand exceeds , and summing over those integers along should give a sum on order , which can be made as large as you like by bringing close to 0.

What’s not so clear to me is: how does

grow with N?

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