Symmetry in Hardy-Littlewood k-tuple conjecture

Let $k in mathbb{N}, k geqslant 2$.

Let $q in mathbb{P}, q geqslant 5 $ and : $$N_q := displaystyle{small prod_{substack{p leqslant q \ text{p prime}}} {normalsize p}}$$ Let : $1 leqslant b leqslant N_q$.

We have : $$gcd(b, N_q) = 1 iff gcd(N_q-b, N_q)=1 tag{1}$$

Then the numbers coprime to $N_q$ and less than $N_q$ are symetric to $dfrac{1}{2}N_q$.

Consider the k-tuple : $mathcal{H}_k := (0,h_1,h_2,cdots,h_{k-1})$, with $0 < h_1 < cdots < h_{k-1}$.

Using $(1)$, if $(b,b+h_1,b+h_2,cdots,b+h_{k-1})$ is coprime to $N_q$ then we have also $(N_q-b-h_{k-1}, N_q-b-h_{k-2}, cdots,N_q-b-h_2, N_q-b-h_1, N_q-b)$ is coprime to $N_q$, (name that property 1).

Consider the k-tuple : $mathcal{H}^{'}_k := (0,(h_{k-1}-h_{k-2}),(h_{k-1}-h_{k-3}),cdots,(h_{k-1}-0))$

Using property1, you can see that : $$b+mathcal{H}_k text{ is coprime to } N_q iff N_q-b-h_{k-1}+mathcal{H}^{'}_k text{ is coprime to } N_q $$

Example : Let $mathcal{H}_3=(0,2,6)$ and $q=7$, for $b=11$ we have $11+(0,2,6)=(11, 13, 17)$ is coprime to $N_7=210$.

We have $N_7-b-h_{k-1}=210-11-6=193$ and $mathcal{H}^{'}_3 = (0, 4, 6)$.

Then we have $193+(0, 4, 6) = (193, 197, 199)$ it coprime too to $N_7$.

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Using Chineese Romander theorem we can prove that : $$#{(b,b+h_1,b+h_2,cdots,b+h_{k-1})inmathbb{N}^{k} , | , 1 leqslant b leqslant N_q , gcd(b, N_q)=gcd(b+h_i, N_q)=1} = displaystyle{small prod_{substack{p leqslant q \ text{p prime}}} {normalsize (p-w(mathcal{H}_k, p))}}$$ Where $w(mathcal{H}_k, p)$ is the number of distinct residues $pmod p$ in $mathcal{H}_k$.

Let $x in mathbb{R}$.

Let $q(x)$ be the largest prime number verifiying $x geqslant displaystyle Big({small prod_{substack{p leqslant q(x) \ text{p prime}}} {normalsize p}}Big)$.

Using prime number theorem we have $q(x) sim log(x)$.

Consider : $$I_{mathcal{H}_k}(x) := #{(b,b+h_1,b+h_2,cdots,b+h_{k-1})inmathbb{N}^k , | , b leq x, gcd(b, N_{q(x)}) = gcd(b+h_i, N_{q(x)})=1 }$$ And : $$pi_{mathcal{H}_k}(x) := #{(p,p+h_1,p+h_2,cdots,p+h_{k-1})inmathbb{P}^k , | , p leq x}$$ We can prove as $x to +infty$ that:

$$I_{mathcal{H}_k}(x) sim mathfrak{S}(mathcal{H}_k) , e^{-gamma k} , dfrac{x}{log(log(x))^k}$$ With $mathfrak{S}(mathcal{H}_k) := displaystyleprod_{text{p prime}}frac{1-frac{w(mathcal{H}_k, p)}{p}}{(1-frac1p)^{k}}$.

If $p in mathbb{P}, p > q(x)=(1+o(1)) log(x)$ then $p$ is coprime to $N_{q(x)}$, this is the relation trivial between prime numbers less than $x$ and numbers coprime to $2,3,cdots,q(x)$ and less than $x$. I give a non-trivial relation as : $$I_{mathcal{H}_k}(x) sim pi_{mathcal{H}_k}(x) big( pi(q(x)) e^{-gamma} big)^k$$ If we prove this conjecture then we have : $$pi_{mathcal{H}_k}(x) sim mathfrak{S}(mathcal{H}_k) dfrac{x}{log(x)^k}.$$ We can find the same results with Goldbach's conjecture or primes of the form $n^2+1$, you can see my article : here