Sorry that I have been bothered with these questions lately. Suppose I know that n/p(n)^a goes to zero, for some a>0, as n goes to infinity. This obviously requires p(n) to diverge. But how slow it can be? If p(n)~n^b for some b>0, then a exists. If p(n)~log(n), can we find such a? My guess is no. Then it brings to my question- how slow the divergence rate of p(n) can be?
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1Do you really mean $a > 0$? You example of $p(n) \sim \log(n)$ is trivially dismissed when $0<a<1$. – Eric Towers Jan 23 '22 at 17:40
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Extremely sorry, Eric. The original version is not clear. I have edited it. – gaussarkov Jan 23 '22 at 17:54
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2Relevant: this answer AND How did Rudin conclude his argument there is no "boundary" between convergent and divergent series? AND Boundary for Series That Converge "Fast Enough" AND Finding a sequence which $\to 0$ "fast enough" AND this 10 September 1999 sci.math post. – Dave L. Renfro Jan 23 '22 at 19:11