need help in proving $\sum_{p^n \leq x, n\geq 2} \frac{\log p} {p} =O(1)$.

Question

This question is a part of merten's formula and I am struck on this particular equation and need help.

Prove that $\sum_{ p^n \leq x, n\geq 2 } \frac{\log p} {p} =O(1)$, where $O$ is big O notation.

I am sorry but I am unable to get any intution, although question seem to be easy and maybe that's why no explanation is given.

A hint would be sufficient.

Thanks.

Answer 1

You've asked to prove that $$ \sum_{p^n \leq x, n \geq 2} \frac{\log p}{p} = O(1). $$ This is false.

The sum is bounded below by $$ \sum_{\substack{p \\ p^2 \leq x}} \frac{1}{p} = \sum_{p \leq \sqrt{x}} \frac{1}{p} \gg \log \log (\sqrt {x}).$$

It is necessary to go about your proof of Merten's formula in a different way.

Answer 2

I think it can be done by induction easily.

1. For n=2 it is more or less clear.

2. Assume it is true for some n.

3. Now when we have n+1, the sum can be divided into two sums - one over $p^{n}\leq x$ (and it is O(1) from induction assumption) and the second one for $p^{n}>x$. Using Prime Number Theorem, how many prime numbers can You have in interval $[p^{n},p^{n+1}]$?