When I've followed a notes that show how obtain a similar asymptotic **using Abel summation formula**, my case with $a_n=\chi(n)$, the characteristic function taking the value 1 if $p$ is prime (in a twin prime-pair, thus **caution** I've defined $\chi(p+2)$ as zero) and $f(x)=x^{\alpha}$, which $\alpha>-1$, and Prime Number Theorem, in my case I am **assuming the Twin prime conjecture**, and **L'Hopital rule** (the author put much careful to write justified computations in the use of L'Hopital rule, I understad all, but he claim that the previous application of L'Hopital rule gives the same result that a more right way, which is *to take an $\epsilon$ and compute the asymptotic limit of the main term with superior limit*, I emphatize other time that the author claims that previous computations are the same using L'Hopital or taking epsilon and computing with superior limits) **applied in my case** $$\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}$$ is asymptotic to $$2C_2\frac{x^{\alpha+1}}{\log^2 x},$$ multiplied by a constant defined precisely by
$$\lim_{x\to\infty}1-\alpha\frac{\int_2^{x}\left(\frac{2C_2t}{\log ^2 t}+o\left(\frac{t}{\log ^2 t}\right)\right)t^{\alpha-1}dt}{2C_2\frac{x^{\alpha+1}}{\log^2 x}}=\frac{1}{1+\alpha}.$$

Thus, when I've used his method I compute for $\alpha>-1$

$$\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}\sim 2C_2\frac{x^{\alpha+1}}{(1+\alpha)\log^2 x},$$ where $C_2$ is the twin prime constant.

Question.Assuming the Twin prime conjecture can you justify rigorously an asymptotic for $\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}$, when $\alpha>-1$? Thanks in advance.

I've defined previous characteristic function and the sum $\sum_{\text{$p,p+2$ twin primes}}p^{\alpha}$, in wich only is added the term $p^{\alpha}$ to follow a similar method corresponding to the author. I don't know if is better add terms $(p+2)^{\alpha}$.