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Let $P$ be the set of primes $p$, $x \geq D \geq z^2 \geq 2$, and let $A⊂[1,x]$ be a set of integers. Suppose $A_{d}=|A| \frac{v(d)}{d}+R_{d}$ for square free d with $v$ being multiplicative and $v(p)$ has an average). Then, according to the Rosser-Iwaniec sieve, $$S(A,P,z) \leq |A|e^{γ}W(z)(1+O(log_{3}D)^{−1})+O(⎜\sum_{d \leq D,μ(d)≠0}{|R_{d}|})⎟$$ where $W(z) = ∏_{p≤z}{(1− \frac{v(p)}{p})}$ and $log_{3}$ is the third iterated logarithm.

Considering primes less that $x$ that can not be found in certain arithmetic progressions modulo primes less than $z$, we have according to the prime number theorem for arithmetic progressions that they are asymptotic to $$\pi(A,p;a=p-2)= Li(x)∏_{p≤z}{(1- \frac{1}{\phi(p)})} + E_{p}$$ where $Li(x)$ is the Logarithm integral which may be approximated by $\frac{x}{log x}$ and $E_{p}$ (the sum of errors involved in our calculations) is guaranteed to be small with respect to the improved Bombieri-Vinogradov theorem for $p < x^{\frac{1}{2}- \epsilon}$ with $\epsilon > 0$. Noting that one can take $\frac{1}{2}- \epsilon$ close to $\frac{1}{2}$ but $\neq \frac{1}{2}$.

Using the result above and the $Rosser-Iwaniec$ sieve result, defining $A$ as $A=\{2<p+2 \leq x:p+2∈P\}$, setting $D = x$ and $z=x^{\frac{1}{2}}$ and taking $W(z) = ∏_{p≤z}{(1− \frac{1}{\phi(p)})} = \frac{2ke^{-γ}}{logz}$, one obtains $$S(A,P,x^{\frac{1}{2}}) \leq \frac{4kLi(x)}{log x}(1+O(log_{3}x)^{−1})+O(⎜\sum_{p \leq x,μ(p)≠0}{|R_{p}|})⎟$$ where $k$ is the "twin prime constant".

Now, my questions are:

(1) What is wrong with this derivation?

(2) If nothing is wrong with it, why is it(something like it) not yet in literature?

I have searched but could not get anything on it and that keeps telling me that something is wrong with the result.

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  • $\begingroup$ @reuns I didn't get what you meant by "It's hard to follow". $\endgroup$
    – Nwik
    Dec 3, 2018 at 7:53
  • $\begingroup$ It's hard to follow because you are trying to refer to complicated general results instead of making things as simple as possible for the special case "counting the twin primes", where $|A_d|,|A|, v(d), R_d$, the $O$ constant can be made explicit. Brun's theorem and brun's sieve are in the litterature, those should be the best possible sieve for twin primes. $\endgroup$
    – reuns
    Dec 3, 2018 at 20:51

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One of your problems is that you are going to find it hard to control the $R_p$ error sum in your final inequality. Calculations of this form were used as long ago as 1973 to establish Chen's improvement on the sum of odd primes problem (my memory is a little fuzzy - I haven't done any number theory for a long time). A reference I used a lot when thinking about this stuff was Halberstam and Richert's Sieve Methods, which may not be in print these days.

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