Symmetry in Hardy-Littlewood k-tuple conjecture

Assuming Hardy-Littlewood $$k$$-tuple conjecture, do the "dual" prime constellations $$(0,h_1, h_2,cdots, h_i,cdots, h_{k-1}=d)$$ and $$(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},cdots,h’_i=h_{k-1}-h_{k-i},cdots,h_{k-1})$$ corresponding to reversed sequences of prime gaps have the same distribution?

If yes, does it imply that the function $$f(n):=dfrac{log g_n}{loglog p_n}$$ and the function $$f'(n)$$ obtained through the substitution $$g_nmapsto g’_n:=dfrac{log^{2} p_n}{g_n}$$ reach the same values an asymptotically equal number of times? Is it related to the functional equation of zeta with which it would then share the same type of symmetry?

(Edited after Lagrida’s answer and accordingly)

MathOverflow Asked by Sylvain JULIEN on February 6, 2021

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 $$N_7$$.

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

Answered by LAGRIDA on February 6, 2021

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