Convergence of a certain series of Primes

Question

This is a problem from Alan Baker's Comprehensive Course in Number Theory. We have to show that $\displaystyle \sum\limits_{p} \frac{1}{p (\log\log p)^{\delta}}$ converges for all $\delta >1$.Here summation is over primes I tried to prove it by considering the series $\displaystyle \sum\limits \frac{1}{n (\log\log n)^{\delta}}$ and showing that it converges. But I was not able to prove it. If anyone can help it would be great. Thanks.

Answer 1

If $p_n$ denotes the $n$th prime, the Prime Number Theorem implies $$p_n (\log \log (p_n))^{\delta} \sim n \log (n) (\log \log (n))^\delta.$$ The convergence of the series follows by comparison.