Non-increasing sequence of positive real numbers with prime index
If $a_n$ is a sequence of non-increasing positive numbers, then suppose we already know that $$\sum_p a_p$$ converges, when $p$ runs over the primes, what should be used to prove that $$\sum_n \frac{a_n}{\log{n}}$$ also converges, where $n$ runs over the positive naturals?
And also, how to show the converse is also true?
Solution 1:
I would use (1) $p_n \approx n\ln n$ and $p_{n+1} < (1+\epsilon)p_n$ for any $\epsilon$ for large enough $n$. This will allow you to handle the step from $a_{p_n}$ to $a_{p_{n+1}}$.