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In a previous question (asked by me), the user who answered the question provide us the asyptotic behaviour of $$\sum_{\text{twin primes }p,p+2\leq x}p^{\alpha},$$ where $\alpha>-1,$ on assumption of an asymptotic involving twin primes. See, if you want all details from the answer here in this Mathematics Stack Exchange.

On the other hand I know that professor Axler studied a problem related to a conjecture due to Mandl (see the first page of [1], is a free access journal).

Question. Let $q_k$ the sequence of twin primes, that is $q_k$ is the $kth$ term of A001359 Lesser of twin primes from The On-Line Encyclopedia of Integer Sequences. I am interested to know if is it possible to set conjecturally (on assumption of a form of the Twin Prime conjecture) a similar inequality than Mandl:

Find a functions $g(x)$ and $h(x)$ such that on assumption of the Twin Prime Conjecture, we can presume that there exists an integer $K_0$ satisfying that $$\sum_{\text{twin primes }q,q+2\leq q_k}q\leq g(k)q_{h(k)},$$ whenever $q_{h(k)}>K_0.$

Many thanks.

Remarks:

1) If you want to add definitions of interesting sequences inspired in your inequality for twin primes, as did Axler inspired in Mandl's inequality feel free to add it.

2) If you can/want to add statistics or plots about our means of twin primes doing a comparison with different functions $g(x)$ and $h(x)$, feel free to do it.

3) I don't know if this question was in the literature, feel free thus to study if you prefer the similar and different problem involving the terms of each twin prime pair $p$ and $p+2$ as summands of our mean $\sum_{\text{twin primes }p,p+2\leq x}p$.

References:

[1] Axler, On a Sequence Involving Prime Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.7.6.

  • All users, here since $h(k)$ is a subscript, then I mean that it is a polynomial with integral coefficients. –  Jul 05 '17 at 20:32
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    What did you try ? – reuns Jul 19 '17 at 13:57
  • Good morning. It is difficult to me dilucidate how start to solve the question. As you see my purpose is to do a comparison with the inequality conjectured by Mandl, now for twin primes. Many thanks @reuns –  Jul 20 '17 at 09:35
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    All we know about $\pi_2(x) = \sum_{p,p+2 \le x} 1$ is that conjecturally $\pi_2(x) \sim C \frac{x}{(\log x)^2}$ where $C$ is the twin-prime constant and that un-conditionally $\pi_2(x) = \mathcal{O}(\frac{x}{(\log x)^2})$. We also know that the same is true for $\pi_{2k}(x) = \sum_{p,p+2k \le x} 1$ and we have some theorems about $\sum_{k \le l} \pi_k(x)$ ie the $k$-twin-prime conjecture averaged on $k$. There are also some better theorems about $\Pi_k(x) = \sum_{pq,pq+2=p'q' \le x} 1$ the twin-semi-prime conjecture. – reuns Jul 20 '17 at 17:15

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