According to my notes it follows immediately from the Prime Number Theorem that we can upper bound a sum over primes with weights $\log p$ by the corresponding sum over integers - for example, $$\sum_{p>x}\frac{\log p}{p^2}\ll\sum_{n>x}\frac{1}{n^2}.$$ However this is not clear to me at all. I am not sure how to change it from a sum over primes to one over integers directly. Could someone please help me with this?
2026-04-01 02:48:41.1775011721
Bounding a weighted sum over primes by a sum over integers
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