I wondered, inspired in a result from [1] (**Proposition 17**) what should be the asymptotic behaviour of the sequence, on assumption of the First Hardy–Littlewood conjecture,


$$\sum\_{\substack{\text{primes }p\leq x\\\text{such that }p+2\text{ is prime}}}\frac{\log^m p}{p}$$


as $x\to\infty$, where $m\geq 1$ denotes a fixed integer. Thus here $p$ denotes the lesser of twin primes (sequence *A001359* from the OEIS) and we assume that there First Hardy–Littlewood conjecture.


A reference for the first Hardy–Littlewood conjecture is this section of [Wikipedia.](https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture)


I don't know if this exercise is in the literature for some fixed integer $m$. I would like to know the deduction for some integer $m\geq 1$.



> 
> **Question.** Deduce for some integer $m\geq 1$ and under the assumption that the First Hardy–Littlewood conjecture is true, what should be the asymptotic behaviour of $$\sum\_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{\log^m p}{p}$$
>  as $x\to\infty$. If it is in the literature, feel free to refer the reference and I try to search and read the result from the literature. **Many thanks.**
> 
> 
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References:
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[1] Christian Axler, *On a Family of Functions Defined Over Sums of Primes*, Journal of Integer Sequences, Volume 22 (2019), Issue 1, Article 19.5.7.



On $\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{(\log p)^m}{p}$, on assumption of the first Hardy–Littlewood conjecture
Using integration by parts, it follows from the [first Hardy-Littlewood conjecture](https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture) that
$$\sum\_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{\log p}{p}\sim 2C\_2\log\log x,$$
and
$$\sum\_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{\log^m p}{p}\sim\frac{2C\_2}{m-1}\log^{m-1} x\qquad\text{when}\qquad m>1.$$


