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I have only two tiny remarks: 1) You can get a sharper bound for $s(n)$ by estimating the sum over $p$, using Brun-Titchmarsh's theorem and integration by parts. This leads to the bound $$s(n) \ll \sum_{d | n} \frac{1}{d \log (2d)} \ll \log (2\omega(n))$$

2)

Sorry I know 1) does not answer your question about the lower bound. It seems to me that the lower bound is quite non-trivial. For example assuming the Generalized Riemann Hypothesis posted something wrong and trying to estimate the sum over $p$ in the obvious way (integration by parts!) didn't lead me anywherecan't delete.

I am going to think more about your question and if I get anywhere I'll post it here!
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I have only two tiny remarks: 1) You can get a sharper bound for $s(n)$ by estimating the sum over $p$, using Brun-Titchmarsh's theorem and integration by parts. This leads to the bound $$s(n) \ll \sum_{d | n} \frac{1}{d \log (2d)} \ll \log (2\omega(n))$$

2) I know 1) does not answer your question about the lower bound. It seems to me that the lower bound is quite non-trivial. For example assuming the Generalized Riemann Hypothesis and trying to estimate the sum over $p$ in the obvious way (integration by parts!) didn't lead me anywhere.

I am going to think more about your question and if I get anywhere I'll post it here!