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Sum of subsequence, over index set of non-zero density, of monotone divergent sum also diver...

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, $$0<\ …
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