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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.

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.

References:

[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.

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Using integration by parts, it follows from the first Hardy-Littlewood conjecture that $$\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{\log p}{p}\sim 2C_2\log\log x,$$ and $$\sum_{\substack{p\leq x\\p,p+2\text{ twin primes}}}\frac{\log^m p}{p}\sim\frac{2C_2}{m-1}\log^{m-1} x\qquad\text{when}\qquad m>1.$$

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    $\begingroup$ It is perfect, many thanks for your hints and answer. $\endgroup$ – user142929 Sep 15 at 8:07

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