Compare two sums using prime number theorem

Question

Let $$A(x)=x\sum_{p \leq x} 1, B(x)=\frac{3}{5}\sum_{x<p\leq 2x}p$$ Using prime number theorem, we have $A(x)\sim\frac{x^2}{\log{x}}$, but how to obtain an estimation for $B(x)$?

Answer 1

The sum of the primes up to $x$ is, asymptotically, $(1/2)x^2/\log x$. So asymptotically, $$B(x)=(3/5)((1/2)(2x)^2/\log(2x)-(1/2)x^2/\log x))$$ Looks like about $(9/10)x^2/\log x$.