Problem 1118 (difficulty: 7/10)

Let \(\displaystyle a_1,a_2,\ldots\) be a sequence of positive reals such that

\(\displaystyle \exists c>0 \quad \forall x>2 \quad \big|\{k:~a_k<x\}\big|>c\frac{x}{\log x}. \)

(Primes for example staisfiy this.) Show that \(\displaystyle \sum\frac1{a_k}=\infty\).

Give me another random problem!