My question is:
Obtain an asymptotic estimate for the sum $$S(x)=\sum_{x<p \leq 2 x} \frac{1}{p}$$ with relative error $1 / \log x$ (i.e., an estimate of the form $$S(x)=f(x)(1+O(1 / \log x))$$ with a simple elementary function $f(x)$.)
I read a (theorem) 4.12 from Apostol's Book.
Theorem 4.12. There is a constant $A$ such that $$\sum_{p \leq x} \frac{1}{p}=\log \log x+A+O\left(\frac{1}{\log x}\right) \text { for all } x \geq 2$$
I tried to construct this theorem again i made some changes but i couldn't find the solution exactly. Can you help me please?