gpt 6 astra pro attempt on Math-59

Hardy-Littlewood Prime k-Tuple Conjecture

Definitions and mathematical statement

For a finite set $H=\{h_1,\ldots,h_k\}\subset{\mathbb{Z}}$ of distinct offsets, let $\nu_p(H)$ be the number of residues it occupies modulo $p$. Admissibility means $\nu_p(H)<p$ for every prime $p$. Define \[ \mathfrak S(H)=\prod_p\frac{1-\nu_p(H)/p}{(1-1/p)^k},\qquad \pi_H(x)=\#\{1\le n\le x:n+h_i\text{ prime for every }i\}. \] For each fixed admissible $H$, the conjecture is $\pi_H(x)\sim\mathfrak S(H)x/(\log x)^k$ as $x\to\infty$. Here $f\sim g$ means $f/g\to1$; this is stronger than infinitude alone.

Formulation and concept references: [S1].

Short English statement

Does every locally admissible fixed pattern of primes occur with exactly the frequency predicted by its divisibility corrections?

Research attempt

Current frontier.

Research check: 27 September 2026. Maynard's multidimensional sieve proves bounded intervals containing many primes, not the asymptotic for each prescribed tuple. Green--Tao's linear-equations framework excludes the parallel linear parts in the single-variable forms $n+h_i$. [E1], [E2]. The 2026 short-interval uniformity theorem in the supplied sources gives Hardy--Littlewood consequences with an additional short average; it does not remove that average for every fixed $H$. [S4]. The primorial proof claim in [S3] is retained as an original source, not accepted as a verified proof. The following argument tests the precise local-to-global passage on which such an approach would have to improve.

Attack route.

We compare three possible routes: direct simultaneous sieving, cancellation in von Mangoldt correlations, and averaging over offsets followed by removal of the average. They respectively require control of a growing sieve modulus, a genuine fixed-shift correlation estimate, or a uniform bound on exceptional offsets. We pursue the first two together, so that one can see exactly which information survives from the local factors.

Attempt: the exact local calculation and its range.

Assume first $k\geq1$ and fix $H$. Write \[ W(z)=\prod_{p\leq z}p,\qquad R_z=\{a\bmod W(z):\gcd(\prod_i(a+h_i),W(z))=1\}. \] The Chinese remainder theorem gives \[ |R_z|=\prod_{p\leq z}(p-\nu_p(H)),\qquad \#\{n\leq x:n\bmod W(z)\in R_z\} =x\prod_{p\leq z}(1-\nu_p(H)/p)+O(|R_z|). \tag{18.1} \] The error follows by counting each residue class as $x/W(z)+O(1)$; its implied constant is absolute. No independence assumption is used. Normalizing the main term by the density for $k$ individually rough integers gives exactly \[ \mathfrak S_z(H)= \prod_{p\leq z}\frac{1-\nu_p(H)/p}{(1-1/p)^k}. \tag{18.2} \] For every sufficiently large prime the offsets are distinct modulo $p$, so $\nu_p(H)=k$. Taylor expansion at zero yields \[ \log\frac{1-k/p}{(1-1/p)^k} =-\frac{k(k-1)}{2p^2}+O_k(p^{-3}). \] The tail converges absolutely; admissibility makes each of the finitely many remaining factors positive. This proves existence and positivity of the singular series, rather than just treating its product formally.

For fixed $z$, $18.1$ is excellent as $x\to\infty$, but it counts composite survivors too. To make roughness equivalent to primality for positive $n+h_i\leq x+\max_i|h_i|$, one would sieve through its square root (apart from the small primes themselves). Formula $18.1$ supplies no useful relative error at that growing modulus. The problem is not the CRT identity: it is the unproved cancellation in its errors when the cutoff is increased with $x$. Taking $x\to\infty$ first at each fixed $z$ does not justify evaluating a diagonal $z=z(x)$.

Attempt: an exact weighted reduction.

Extend $\Lambda(t)$ by zero for integers $t\leq0$, and put \[ C_H(x)=\sum_{1\leq n\leq x}\prod_{i=1}^k\Lambda(n+h_i). \tag{18.3} \] A precise sufficient statement is $C_H(x)=\mathfrak S(H)x+o_H(x)$ for every fixed admissible $H$. Here is the removal of both prime powers and logarithmic weights. The number of $n\leq x$ for which some $n+h_i=p^a$ with $a\geq2$ is $O_H(\sqrt{x}\log x)$, by summing the elementary bounds $(x+O_H(1))^{1/a}$ over $2\leq a\leq\log_2(x+O_H(1))$. Every term in $18.3$ is $O_H((\log x)^k)$. Therefore the total contribution from such powers is $o_H(x)$.

Let $A_H(t)$ count genuine prime tuples up to $t$ and let $w(t)=\prod_i\log(t+h_i)$ for $t$ larger than a fixed constant $t_0(H)$. The weighted count $B_H(x)=\int_{t_0}^xw(t)\,dA_H(t)$ consequently satisfies $B_H(x)=\mathfrak S(H)x+o_H(x)$. For large $t$, \[ w(t)\sim(\log t)^k,\qquad \frac{w'(t)}{w(t)^2}=O_H\!\left(\frac1{t(\log t)^{k+1}}\right). \] Stieltjes integration by parts gives \[ A_H(x)=\frac{B_H(x)}{w(x)} +\int_{t_0}^x B_H(t)\frac{w'(t)}{w(t)^2}\,dt+O_H(1). \tag{18.4} \] Since $B_H(t)=O_H(t)$, the integral is $O_H(x/(\log x)^{k+1})$; one can bound it by splitting at $\sqrt{x}$. Thus $18.3$'s proposed estimate implies exactly the stated unweighted asymptotic, with no extra uniformity in $H$ demanded. Conversely the unweighted asymptotic and integration by parts recover $18.3$, after the same prime-power estimate. This is an equivalent analytic target, not a new proof of it. Negative offsets affect only finitely many initial terms. If the empty tuple is admitted, its assertion is simply $\lfloor x\rfloor \sim x$ and needs no argument.

Attempt: what an averaged theorem cannot by itself remove.

The obstacle can be made quantitative without pretending to build fake primes. For any positive comparison constants $s_h$, let nonnegative model counts satisfy $b_h(X)=s_hX$ except for one fixed $h_0$, where $b_{h_0}(X)=0$. Then \[ \frac1H\sum_{h=1}^H \left|\frac{b_h(X)}{s_hX}-1\right|=\frac1H \quad(H\geq h_0). \tag{18.5} \] Perfectly good mean convergence with $H\to\infty$ coexists with a permanently bad fixed offset. This countermodel invalidates removal of an average solely by nonnegativity. It is not a model of the primes and not a counterexample to Hardy--Littlewood.

One possible strengthening would be a uniform estimate $|C_H(x)-\mathfrak S(H)x|\leq \epsilon(x)x$ for every admissible tuple in a box whose radius tends to infinity, with $\epsilon(x)\to0$ and the number of offsets fixed. Every fixed tuple would eventually be included. But that strengthening already contains the fixed-shift difficulty; the current averaged statements do not supply it. A different route would prove $18.3$ by combining the CRT factors with cancellation beyond the sieve's parity obstruction, rather than identifying survivors with primes.

Stress test.

The convergent singular series is only a local density correction. The error in $18.1$ is not uniform enough for primality detection. No argument here promotes higher uniformity in a finite-complexity system to the parallel-shift system without checking its hypotheses. The models in $18.5$ test an inference, not arithmetic reality. Proving $18.3$ even for $H=\{0,2\}$ would yield the twin-prime assertion of excluded UnsolvedMath problem 11; that dependency is explicitly retained, although no separate file for UnsolvedMath problem 11 is produced.

Outcome.

Outcome: Exploratory approach with unresolved gap. The exact CRT count and weighted/unweighted equivalence isolate the missing fixed-tuple asymptotic $18.3$. Neither the sieve error nor the exceptional-offset error is controlled at the required scale. No novelty is claimed for the classical ingredients.

Sources

• S1 k-Tuple Conjecture (MathWorld). https://mathworld.wolfram.com/k-TupleConjecture.html. Catalogue formulation source.
• S2 Correlations of almost primes — Institut Henri Poincaré seminar abstract. https://www.ihp.fr/en/node/3096. Catalogue research/status reference; not a proof endorsement.
• S3 Proof of the Hardy-Littlewood K-tuple Conjecture in the Distribution of Numbers Coprime with the Primorial — Tim Samshuijzen, May 2026 revision. https://rxiv.org/pdf/2501.0157v4.pdf. Catalogue research/status reference; not a proof endorsement.
• S4 Higher uniformity of arithmetic functions in short intervals II. Almost all intervals — Matomäki, Radziwiłł, Shao, Tao and Teräväinen $2026$. https://link.springer.com/article/10.1007/s00222-026-01408-6. Catalogue research/status reference; not a proof endorsement.
• E1 James Maynard, Small gaps between primes, arXiv:1311.4600 $2013$, Theorem 1.1 and the sieve setup. https://arxiv.org/abs/1311.4600. Added research source. Bounded clusters versus a prescribed simultaneous prime pattern. Consulted 27 September 2026.
• E2 Ben Green and Terence Tao, Linear Equations in Primes, arXiv:math/0606088, introduction and finite-complexity hypotheses. https://arxiv.org/abs/math/0606088. Added research source. Scope of the linear-forms results; parallel shifts are not covered by the finite-complexity case. Consulted 27 September 2026.