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 $(p+2)^{\alpha}$.
This can be done using partial summation in a way that is similar to this answer: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?
Let $\pi_2(x)=\sum_{\text{twin primes }p,p+2\leq x}1$. Then $$\sum_{\text{twin primes }p,p+2\leq x}p^{\alpha}=\int_1^x t^{\alpha}d\pi_2(t).$$ Assuming that $$\pi_2(x)\sim 2C_2\int_2^x \frac{1}{(\log t)^2}dt,$$ by properly rearranging to control the error term as was done in that previous answer, we find that $$\sum_{\text{twin primes }p,p+2\leq x}p^{\alpha}\sim 2C_2 \int_1^x \frac{t^\alpha}{(\log t)^2}dt$$ and by setting $t=u^{1/(1+\alpha)}$ we have $$\sum_{\text{twin primes }p,p+2\leq x}p^{\alpha}\sim \frac{2C_2x^{1+\alpha}}{(1+\alpha)^2(\log x)^2}.$$