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Is there a "slowest possible" increasing finite function $h(n)$ such that the modified harmonic series $$S_h = \sum_{n=1}^\infty \frac{1}{n\cdot h(n)}$$ converges? "Slowest possible $h$" here I think has the sense that whenever $S_f$ converges for some $f$ then $h = O(f)$.

Note that the requirement for $h$ to be an increasing finite function excludes things like $\sum_p 1/(p\ln p)$ which converges by not summing over all integers.

Luke Pritchett
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    You can adapt answer https://math.stackexchange.com/a/1060619/659499 to your seires - essentially, you can always divide $h(n)$ by something growing slow enough but still going to $\infty$, while preserving convergence. – mihaild Feb 04 '25 at 16:48

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