claims depot shelf
The Twin Prime Conjecture
Formal claims (Lean)
Stated claims
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The central claim is that short arithmetic progressions cannot cover $1,\dots,n$ when the logarithm of their largest term is much smaller than $\sqrt n$. Formally, any $n$-divisor progression with $\log H = o(\sqrt n)$ has $L \ge (\sqrt{8/27} - o(1)) n^{3/4}/\sqrt{\log n}$. A direct corollary excludes every exponent pair $\alpha<1/2$, $\beta<3/8$, including the originally proposed $(1/3,1/3)$, even under $n^{o(1)}$ slack. The obstruction is sharp at the level of the argument: the constant $\sqrt{8/27}$ arises from optimizing the band $a=2/3$, $b=1$ below $\sqrt n$ and is not claimed to be optimal.
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Conjecture (S) — every even n ≥ 6 can be written as p + t with p prime and t a twin member — sits strictly between the Goldbach and Dubner conjectures. The paper's central discovery is twofold. First, (S) is a single elementary statement that implies both the Goldbach conjecture and the twin prime conjecture (Theorem 1). Second, any proof of (S) would have to contend with a mod-3 orientation rigidity: for even n not divisible by 3, every twin member appearing in a Goldbach partition of n is a lower member (t+2 prime) when n ≡ 1 mod 3, and an upper member (t−2 prime) when n ≡ 2 mod 3, with the only exceptions being the primes 3 and 5. The paper verifies (S) exhaustively to 10^12, computing th
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The central claim is that the three Mertens sums satisfy explicit inequalities with the same shape as the best current Chebyshev-function bounds. In particular, Theorem 2 proves $|\lambda(x)-\log\log x-M|\le A_\lambda(x_0)(\log x)^{1/2}\exp(-C\sqrt{\log x})$ for every $x\ge x_0$, with $A_\lambda(2)=9.2203$ and $C=0.84768$, and also proves $|\lambda(x)-\log\log x-M|\le A_\ell(x_0)/(\log x)^\ell$ for $\ell=1,\dots,5$. Theorems 3 and 4 give matching bounds for $\Upsilon(x)=\sum_{p\le x}(\log p)/p$ and $\tilde\psi(x)=\sum_{n\le x}\Lambda(n)/n$, with $A_\Upsilon(2)=A_{\tilde\psi}(2)=9.2203$. From these bounds the paper derives two-sided inequalities for the Mertens products $\prod_{p\le x}(1-1/p)
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The paper's central discovery is a rigidity statement at the level of the measure \(\Lambda_{f,t} = \sum_p \frac{f(p)}{p}\delta_{\log\log p - t}\): if \(A_1(X_t;f)\to\kappa\) for \(X_t=\exp(e^t)\), then \(\Lambda_{f,t}\to\kappa\, du\) vaguely as \(t\to\infty\). This is shown by writing \(A_1\) and \(A_2\) as convolutions with the Dickman kernels \(K_1(u)=\rho(e^u-1)\) and \(K_2(u)=\$int_1^{{e^u-1}}$\rho(e^u-1-v)\frac{dv}{v}\), up to \(o(1)\). The convolution identity for \(A_1\) and the nonvanishing of \(\widehat{K_1}\) on the real line imply, by Wiener's Tauberian theorem, that every subsequential limit of \(\Lambda_{f,t}\) is the constant \(\kappa\) times Lebesgue measure. The convolution identity for \(A_2\), together with \(\int K_2 = 1\), then gives \(A_2(X_t;f)\to\kappa\).
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The core claim is that the least quadratic residue modulo n has exponential order 4^k in the number k of odd prime factors, and that this phenomenon is stable enough to transfer to binary quadratic forms. Theorem 1.1 supplies the two sides: a pigeonhole argument shows ℓ(n) ≤ C k² 4^k, while a finite-field construction, translating primes into irreducible polynomials over F₂[x], produces moduli n for which any r with ℓ(n)=r must carry more than 2^k distinct algebraic roots, forcing log r ≥ k log 4 − O(k/log k). Theorem 1.2 sharpens this into a discriminant statement: for any Δ one can build an odd square-free n with log n ≤ C log Δ (log log Δ)² (and under GRH log n ≤ C log Δ log log Δ) such t
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On the paper's own terms, the central discovery is a quantitative lower bound: for sufficiently large n, h_P(n) ≥ n exp((1/50) log n / log log n), and therefore the same inequality holds for F(n). The proof constructs, for a carefully chosen product W of small odd primes, a digit-product set A_t that contains a k-term arithmetic progression of every difference d < W^t (a (k,W^t) arithmetic-Kakeya set) while having size at most W^{t(1−ε)}. Embedding the primes in a CRT-chosen interval then makes the neighborhood of the prime set too small for a matching, so the interval must be long. A separate argument, based on a projection estimate for finite lattice sets, yields F(n) ≤ n^{1.4031+o(1)}; an
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The paper proves that W(u) equals 2|Φ_2(u)|cos(arg Φ_2(u)) plus an explicit error θ_2(u), where Φ_2(u)=Φ(u)(1+α(u)), Φ(u)=exp{−uζ+J(−ζ)} divided by the square root of 2πu(1−1/ζ), J(−ζ) is the incomplete gamma function Γ(0,−ζ), and ζ(u) is the unique solution of e^ζ=−uζ in a specified region, equivalently the 1-branch of the Lambert W function applied to 1/u. The error is bounded by 420 u^{−6} for 6≤u≤19 and by 0.005 u^{−2} for u≥16; a simpler corollary gives |W(u)|<2|Φ(u)| for u≥3. The same method yields a refinement with error below 0.01 u^{−3} on 18≤u≤1000 and below 8.4 u^{−3} for larger u. This is achieved by the saddle point method: the Laplace transform of ω is e^{J(s)}−1, the contour i
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The central claim is Theorem 1.1: for every nonzero f(x) ∈ Z[x], the set {n > 0 : p_n ∤ f(n)} is infinite. The paper shows this by contradiction: assuming p_n | f(n) for all large n, it uses the bounded-gaps theorem to find infinitely many n for which a fixed block of k+1 consecutive primes has fixed offsets h_0,...,h_k. It then constructs a nonzero polynomial Φ(x,y) with integer coefficients such that, on that infinite set, Φ(n,p_n)/ρ(p_n) is an integer that is also o(1), forcing Φ(n,p_n)=0 eventually, which contradicts the leading-term growth of Φ. Since the existence of such f would mean π(p) satisfies a nontrivial polynomial relation over Q, the conclusion is that π(p) ∉ C_A, the integra
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Theorem 1.4 asserts that as x→∞, the count of n<x with P^+(n)<P^+(n+1) exceeds 0.280x, and by symmetry the same holds for the reverse inequality. The constant comes from a decomposition of n according to the sizes of P^+(n) and P^+(n+1): a cutoff c=0.1348 separates small and large cases, and each piece is estimated either by the standard smooth-number distribution or by a new lower-bound function C(η) for smooth numbers in arithmetic progressions. With the choices δ1=0.417 and an optimized auxiliary function t1(η), the assembled lower bound C(c,δ1)−2ν is reported to exceed 0.280.
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The paper's central claim is that the Skolem Problem reduces to a question about prime gaps. It defines good primes as those that are not 'bad', where badness means that a prime divides the algebraic norm of a certain exponential-polynomial expression associated to a small LRS around a potential large zero. For any large zero of an LRS, the paper shows there must be an interval around the zero containing far fewer good primes than the prime-gap conjecture would predict; if the conjecture holds for good primes, this contradiction rules out large zeros. The unconditional theorem shows that across all LRS, the set of large-zero indices is so sparse that its complement still allows an algorithm
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For every digit d from 0 to 9, the total number of occurrences of d among all digits of all primes less than 10^n, divided by the total number of such digits, equals 1/10 plus an error of size O((log n)/n) as n tends to infinity, uniformly in d. The same limit therefore holds in total variation for the empirical digit measure.
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For beta-sieve weights (or suitable Lambda-squared weights) the sum U = sum_d phi(d) (sum_{m: d|m} lambda_m/m)^2 is << product_p (1-1/p). This is the same order of magnitude previously obtained for the larger sum W, and is precisely the quantity needed for the short-interval applications.
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For each r > 2 there exist constants r' < c(r) < 2 < C(r) with lim c(r) = 1 and lim C(r) = ∞ such that the r-variation of the prime-weighted Carleson series is bounded on ℓ^p for all c(r) < p < C(r), while the variation is unbounded for p ≤ r'. The same reductions give that the maximal prime Carleson operator is bounded on ℓ^p(ℤ) for every 1 < p < ∞.
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Guth and Maynard prove a new large-values estimate for Dirichlet polynomials that yields the zero-density bound N(σ,T) ≪ T^{15(1-σ)/(3+5σ)+ε} uniformly for 1/2 ≤ σ ≤ 1; this is the first improvement on Ingham's 1940 estimate throughout the range σ ≤ 3/4 and, when combined with earlier results, gives A(σ) < 30/13, which in turn implies the prime-number theorem in short intervals of length x^{17/30} and almost all intervals of length x^{2/15}.
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For every fixed k≥2 and every N≥0 the additive Mertens sum admits the complete expansion Sk(x)=(x^{k-1}/log^k x)∑_{n=0}^N Ek,n/log^n x + O(x^{k-1}/log^{k+N+1}x), where the coefficients are the absolutely convergent integrals Ek,n=(-1)^n ∫_{(0,1]^k} hn(log t1,…,log tk)/(t1+…+tk) dt and hn is the complete homogeneous symmetric polynomial of degree n.
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Assuming the Riemann Hypothesis and a mild lower bound on the Gaussian smoothing width relative to a fixed simple critical-line zero, the full Gaussian–Perron prime-force defect near that zero equals the universal selected-zero profile −a Re(e^{−λ}/λ) plus an error that is O(1/log X) plus exponentially small, uniformly on compact sets of the logarithmic displacement λ away from zero.
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On the paper's own terms, the central discovery is that Gilbreath's conjecture is equivalent in effect to a balance property of the right diagonal of the absolute-difference triangle. Once the triangle's rightmost column reaches a long enough stretch of 0s and 2s, any subsequent term that is not too large keeps the bottom cell equal to 1. Theorem 5.5 makes this quantitative: with g*_n < n^α and ν2(q_{n−1}) > n^β and β>α, a valid successful sequence cannot fail at the next step. Applying the known prime-gap exponent α=0.525 (Baker 2001) and conjecturing β=0.99 for the primes, the author obtains what he describes as a trivial proof of Gilbreath's conjecture if the conjecture holds, and offers
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For a Serre curve — an elliptic curve over Q whose adelic Galois image is as large as possible, namely the index-2 subgroup forced by the quadratic field Q(√Δ_E) — Theorem 3 provides a closed formula for C^{m-div}_E, the natural density of good primes p with m | #E_p(F_p). Writing m = m_1 m_2 with m_1 = gcd(m, m_E^∞), the density equals the average density C^{m-div} when m_1 is outside a small exceptional 2-adic range; otherwise it is (C^{m1-div} + ∏_{ℓ^α∥m1} −ℓ^{2α−1}/|GL_2(Z/ℓ^αZ)|) · C^{m2-div}. This formula yields Corollary 4: C^{m-div}_E > 1/m for every m ≥ 2. Theorem 5 then shows that the k-th power average of |C^{m-div}_E − C^{m-div}| over the box family tends to zero as the box grows
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The main result, Theorem 2.2, states that if n=pq and n+12=p'q' are both products of two distinct primes with p<q and p'<q', then the two jump conditions φ(n+12)-φ(n)=12 and σ(n+12)-σ(n)=12 hold together if and only if p≥5, q=p+8, and {p',q'}={p+2,p+6}. The proof reduces both equations to the equality p+q=p'+q', then uses the discriminant relation (q-p)^2-(q'-p')^2=48. Factoring 48 gives only three possible pairs of square gaps; parity rules out all but (8,4), forcing the prime-quadruplet spacing. A separate theorem shows no prime power p^k with k≥2 can satisfy the first jump condition alone.
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We exhibit an explicit integer M < 1.78 × 10^{32} such that every integer m ≥ M with gcd(m,9)=1 occurs as s(p) for at least one prime p. The proof uses explicit major-arc estimates, a fully explicit replacement for the implicit prime exponential-sum input, and constant-tracked Type-II minor-arc estimates.
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We prove unconditionally that Proposition (1+1.9) is true. Thus every sufficiently large even integer N can be written as N = p + r q, where r ≤ q^{0.9}, r is 1 or prime, and p, q are primes. Assuming the Elliott-Halberstam Conjecture the exponent 1.9 can be replaced by 1.4. The same method yields Proposition (1-1.75) for the twin-prime side unconditionally and (1-1.4) under Elliott-Halberstam.
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For every k ≥ 1 the number of primes P_k in the interval I_k between consecutive odd squares equals N_k − S_k + E_k, where N_k = 4k is the number of odd integers in I_k, S_k is the total matrix multiplicity of those integers, and E_k corrects for excess multiplicity of non-semiprime composites. Consequently a prime exists in I_k if and only if the combinatorial inequality E_k ≤ S_k − N_k holds.
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On the paper's own terms, the central discovery is Theorem 2.5: g_n < (51/16) log² p_n for every n ≥ 1. The engine is a set of summation identities (Theorem 2.1) expressing each gap g_n through B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1, where A_k is the average of the first k gaps. Lemma 2.2 asserts that B_n < 2B_{n−1} for every n ≥ 22; Lemma 2.3 converts that into g_n < (51/16) B_{n+1}; and explicit classical bounds on the nth prime, n log n < p_n < n(log n + log log n), let the author bound B_{n+1} by a logarithm-squared expression. The cases n ≤ 21 are checked directly.
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The positivity of Cojocaru's constant C_{E,j} — which controls the density of primes where the smallest invariant factor of E(F_p) is j — can be studied through the image of the adelic Galois representation of E, and experimentally the constant vanishes precisely when division fields of E coincide. This reframes an analytic density question as a group-theoretic and field-arithmetic one, and the authors classify several coincidence families arising from abelian division fields.
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In the paper's own terms, the discovery is that Chebyshev's bounds follow from comparing two weighted prime sums: π(x) ≍ (∑_{p≤x}√(ln p/p))² − ∑_{p≤x} ln p/p, and (∑_{p≤x}√(ln p/p))² ≍ x/ln x. The second statement is shown by combining Abel's identity with Mertens' estimate; the first is shown by writing the difference E(x) at prime indices and using the exact recurrence E_n = E_{n-1} + 2a_n S_{n-1}, where a_n = √(ln p_n/p_n). Since S_n ≍ 1/a_n, each increment 2a_n S_{n-1} is bounded above and below by positive constants, forcing E_n ≍ n = π(p_n). Thus the prime count is sandwiched twice, yielding the Chebyshev bounds.
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The paper establishes an asymptotic formula for the average of the error term appearing in Bateman–Horn’s conjecture, taken over an exponential range of arguments; the resulting main term is strictly larger than square-root size, so the averaged error exhibits worse-than-square-root cancellation.
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On the paper's own terms: Mersenne prime exponents have elevated divisor complexity of p-1 relative to local prime controls, and this elevation survives matching by exponent size. The mean percentile rank of S(p) in a 5000-prime window is 0.648 versus a null expectation of 0.5; sign, Wilcoxon, and permutation tests reject the null with p-values around 0.002. The paper interprets this as evidence for a structural refinement of the standard probability estimate for M_p being prime, preserving the log p / p scale after averaging over S.
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Informational cardinality I(M)=(α(M),δ(M),ι(M)) distinguishes the essential fractal prime set P_ess from the generalized Cantor set C_{1/3}: both have continuum cardinality (α=1), yet dim_H(P_ess)=1/2>1/3=dim_H(C_{1/3}) and ι(P_ess)=-ζ(1/2) while ι(C_{1/3})=0, so I(P_ess)>I(C_{1/3}) under lexicographic order. The same framework pairs P_ess with a fractal zero set Z_F of equal dimension and conjectures that their information measures sum to zero.
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The central discovery is that a deterministic fractal set, PF, can be built directly from the prime residues modulo 16 and that its fractal dimension is exactly 1/4, with Hausdorff and box-counting dimensions agreeing. At each stage, every interval is split into 16 equal pieces and only two are kept, those indexed by the current prime residue a_n and its opposite a_n+8 mod 16. The proof shows that the number of length-16^{-n} intervals needed to cover PF is exactly 2^n, giving box dimension log 2/log 16. For the Hausdorff dimension, the paper constructs a natural mass distribution and proves a measure-decay estimate, then uses the mass distribution principle to get the lower bound; the upper
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On the author's own terms, the paper establishes that primes have an average distribution in arithmetic progressions compatible with moduli beyond sqrt(x). In particular, Theorem 1.1 asserts that for composite moduli q = q1 q2 with Q1^2 Q2 < x^{1-epsilon} and Q1^7 Q2^12 < x^{4-epsilon}, any divisor-bounded bilinear weight gives a discrepancy sum of size O(x/(log x)^A). Since those two constraints are compatible up to q1 q2 = x^{9/17}, this is a bilinear mean-value theorem at modulus size x^{9/17}; the analogous trilinear statement reaches x^{17/32}. The paper also constructs explicit functions rho0(n) <= 1_p(n) <= rho1(n) whose sums and arithmetic-progression distributions are controlled, an
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Theorems 1.1 and 1.2 assert that for h = X^θ with 2/15+ε < θ < 0.99, all but O_ε(X exp(-c_ε (log X)^{1/4})) integers x ∈ [X,2X] satisfy R(x) ≍ h and S(x) ≍ h, where R(x) counts representations of n ∈ (x,x+h] as p+a (p prime, a in a lacunary set A_λ(X) of size ≍ log X) and S(x) counts distinct such n. The lower bound R(x) ≫ h comes from a quoted almost-all primes-in-short-intervals asymptotic (restricting to a ≤ h/2); the upper bound on the second moment Q(x) ≪ h comes from an upper-bound sieve for prime pairs plus a structural estimate on small prime factors in differences a_1-a_2; Cauchy–Schwarz then gives S(x) ≥ R(x)²/Q(x) ≫ h.
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On the paper's own terms: a proposition P is α-decidable if there exists a Brouwer ordinal y such that P ↔ y ≥ α. Using the type Brw of Brouwer trees, whose limit constructor accepts only strictly increasing sequences, the paper shows that 1-decidability coincides with ordinary decidability and (ω+1)-decidability with semidecidability. The main theorem (Theorem 7.5) states that for a family P : N → Prop of semidecidable propositions, the countable meet ∀n.Pn is ω²-decidable. The proof constructs a characteristic ordinal Ψ(P) for such a family and establishes the equivalence ∀i.Pi ↔ Ψ(P) ≥ ω², relying on Lemma 7.7, which says that dropping a finite prefix does not push the characteristic ordi
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The central claim is Theorem 1.4: along the sequence of primes p_n, the return-times averages converge pointwise for all second systems, and the Wiener–Wintner averages converge for each fixed frequency θ. The mechanism is to write the prime indicator through the von Mangoldt function, approximate it by a truncated Heath–Brown model, control the U^3 norm of the part of the model with fixed rational complexity Q by the bound Q^{-3/8+o(1)}, and then transfer Gowers-norm control of the weight to almost-everywhere convergence of weighted ergodic averages via uniform Wiener–Wintner and weak-mixing reductions.
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The load-bearing result is Theorem 1: for all 2≤n≤4,553,432,387, every row of the Sierpiński matrix S_n contains at least one prime. The proof works because the largest prime gap starting below (4,553,432,387)² is the 82nd maximal gap, of width 1,572, so the pigeonhole principle guarantees at least floor(n/1,572) primes in each row for n in that range. The small cases n≤13 are checked directly, or by a known 1961 verification used to cover the overlap n≤4,505; the two ranges splice together to cover all n up to N₈₃. For n beyond N₈₃, Theorems 3 and 7 provide guarantees that at least n/4 rows, and the bottom 131,294 rows, contain primes.
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If α is an irrational number having a continued fraction expansion with bounded terms, then the number of primes p in the interval (X-Y,X] satisfying ||pα||<δ is asymptotically equal to 2δ Y/log X, provided that X≥10, X^{2/3+ε}≤Y≤X/2 and X^ε max{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}}≤δ≤1/2.
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The central claim is that a scaled omega-Gibbs prior on prime multipliers, pi_p proportional to 2^{-beta * ell_omega(p)} with beta > 1, makes the MTE tractable and universal: additive gaps X_{t+1} - X_t have asymptotically Pareto tails with exponent beta, and the time-averaged omega codelength converges almost surely. The pure beta = 1 prior is a critical boundary case: it is regularly varying with index 1, has infinite first moment, and lies outside the theorems. On Debian and PyPI package-size data, the fitted slope a is below log 2, implying beta < 1, which the authors read as the signature of human-generated, non-averaging complexity.
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Barrett's formula asserts that Barr(n) = 3 + sum_{k=5}^{n-1} floor( sin(pi (k-1)!/k) / sin(pi/k) ) equals the number of primes p < n, under the historical convention that 1 is counted as prime. The summand is a geometric indicator of primality: for composite k ≥ 5, (k-1)! is divisible by k, so the numerator's sine is zero; for prime k, Wilson's theorem forces the ratio to be 1. Thus every integer k contributes exactly 1 if k is prime and 0 otherwise, making the formula a finite, exact trigonometric expression for the prime-counting function.
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The paper's core discovery is Theorem 8.2: RH holds if and only if the trajectory error functional satisfies E(X) ≪ X^(1/2) log X for all X ≥ e^120. The proof runs through a chain of unconditional estimates—one-visit and parent-window lemmas limiting how often a trajectory hits a window, macro-step alignment showing that L ≍ log X composite steps contract the scale from X to X^(3/4), and a frequency-netting lemma that controls the sum over Riemann-zero contributions using an explicit smoothed formula with cubic-log truncation and a grid-based large sieve. Iterating the contraction inequalities yields the unconditional bound E(X) ≪ X^(1/2) log X (Corollary 6.6), and the paper then argues that
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Theorem 2.6 of the paper is that the sliding-window 'puzzle' algorithm computes h(k,x) on input x in O(x log x) arithmetic operations and O(x^(2/3)) space. The priority-queue algorithm, which keeps one chain of each length in a min-heap keyed by sum, achieves O(x log x) time and O(x^(3/5)) space when primes are produced by an incremental interval sieve, or O(x(log x)^2) time and O(sqrt(x log x)) space when primes are produced by a pseudosquares prime test. The analysis is average-case over prime gaps: the basic estimate, Lemma 2.3, is that sliding a chain of length m with sum n changes the sum by Theta(m log(n/m)) on average. The implementations ran to x = 10^14 + 145300 (the paper notes tha
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The paper's central claim is that old existence theorems for primes in short intervals can be made fully explicit by splicing together three independent effective inputs. Using a cited 90th-power theorem, interpreted by inspection as [x, x + 90 x^(1 - 1/90)] for all real x >= 1, plus the cited cube theorem with its improved threshold, plus the known table of maximal prime gaps and explicit linear interval data, the paper derives three effective statements: for n >= 4 and x >= exp(exp(33)) the interval [x, x + x^(1 - 1/n)] contains a prime; for n >= 91 and x >= [90^90]^(n/(n-90)) it contains a prime; and for n >= 106 the statement holds for every x >= 1. The last statement yields the universa
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The central claim is that pointwise ergodic averages along the set of polynomial values of primes of the form x^2 + n y^2 converge almost everywhere for every function in L^p with p > 1, for every measure-preserving system. The proof adapts the Hardy-Littlewood circle method to this arithmetic setting, with the novel ingredient being estimates for the distribution of the prime ideals that correspond to the quadratic form. The paper further constructs an L^1 function for which these averages fail to converge, proving that the p > 1 condition is optimal.
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The central claim is that for an irrational $\alpha$ and fixed $\tau\in(0,1/8)$, the set $\mathcal{P}_\tau(\alpha)=\{p\text{ prime}:\lVert\alpha p\rVert\le p^{-\tau}\}$ contains infinitely many nontrivial three-term arithmetic progressions. The novelty is that membership depends on a single irrational multiplier and the selected primes become sparser as $p$ grows, yet additive configurations of length three still occur infinitely often. The paper further states a binary Goldbach-type result for sums of two primes from the same thin Bohr set.
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The core discovery is a finite counterexample: the analogue of the second Hardy-Littlewood conjecture for cyclic numbers is false. In its prime form, that conjecture says the number of primes in an interval $(x,x+y]$ never exceeds the number of primes up to $y$. Replacing primes by cyclic numbers gives the inequality $C(x+y)-C(x)\le C(y)$, where $C$ counts cyclic numbers; the paper reports that this inequality fails somewhere among cyclic numbers below $10^8$. The enumeration total is $28{,}488{,}167$, and the equivalence $n$ cyclic $\iff \gcd(n,\phi(n))=1$ is what makes the search purely arithmetic.
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The prediction is that in almost all prime gaps $(p_n,p_{n+1})$, there is a natural number $m$ with least prime factor $p(m) \ge p_{n+1}-p_n$. The paper establishes this prediction. Let $N(X)$ count exceptional $n$ with $p_n \in [X,2X]$, meaning no such $m$ lies in the gap. The main unconditional result is $N(X) = O(X/\log^2 X)$. Conditionally on a form of the prime tuples conjecture, the paper obtains $N(X) \sim cX/\log^2 X$ for an explicit constant $c>0$ numerically suggested to be about $2.7$--$2.8$. The proof is a sieve-theoretic argument; the full-strength version depends on a previously developed asymptotic for singular series.
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The central claim is that $f_{Z,k,H}(n)=n^2-(2Zk-1)n+\frac{(2Zk-1)^2+H}{4}$ is a prime-generating quadratic whose behavior can be controlled by the pair $(Z,k)$ while $H$ ranges over the Heegner numbers. The paper states that through specific substitutions this form is closely related to the Euler-Rabinowitsch polynomials, so it both recovers known prime-rich quadratics and adds extra tuning freedom. Using the Bateman-Horn conjecture and prime-counting functions, the paper asserts that the family can be optimized to generate a high density of primes, and takes this as evidence that Heegner numbers have a concrete impact on the distribution of primes in quadratic sequences.
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The central claim is that $N(\sigma,T)$ is $O(1)$ uniformly in $T$ when $\sigma$ is sufficiently close to $1-\nu(t)$, the left edge of the Korobov-Vinogradov zero-free region, where $\nu(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$. Exploiting this uniform boundedness in the minimization that defines $\omega(x)$, the paper obtains $\psi(x)-x \ll x\exp\{-\omega(x)\}$ with no $\varepsilon$ slack, i.e. $\varepsilon=0$ in the stated bound.
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The central claim is that two perturbations of Maynard's sieve raise the optimal ratio $M(F)=I(F)/J(F)$ by an additive amount $\delta/2 + \epsilon\ln\ln k$ beyond the baseline $\frac{1}{4}\ln k$. The first perturbation enlarges the simplex $R=\{t\in[0,\tau]^k:\sum t_i\le 1\}$ to $R'=\{t\in[0,\tau]^k:\sum t_i\le 1+\delta\chi(\sum t_i)\}$, where $\chi$ is a logistic-map iterate, and the second replaces $F$ by $F+\epsilon\xi$ with $\xi(t)=\prod_{j=1}^k\Phi(t_j)$ built from the normal CDF as a stand-in for GUE spacing statistics. Theorem 4.3 encodes the resulting heuristic formula $M'(F')\sim\frac{1}{4}\ln k+\frac{\delta}{2}+\epsilon\ln\ln k$. The paper then uses the standard sieve criterion $M>m \Rightarrow$ infinitely many $m+1$ primes in an admissible $k$-tuple to claim $M'>3$ at $k\approx 40$, yielding 4-prime blocks and hence gaps of at most 180 unconditionally and at most 8 under a partial Elliott–Halberstam assumption with $\theta=0.8$.
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Theorem 1.10: For any function φ : N → R_{>0} and ℓ ∈ N, we have dim_H E'_ℓ(φ) = dim_H E_ℓ(φ). If correct, this means restricting infinitely many partial quotients to be prime does not change the Hausdorff dimension of the set where products of ℓ consecutive partial quotients exceed φ(n). The paper also claims the zero-one law in Theorem 1.8: m(E'_ℓ(φ)) = 0 or 1 according to the convergence or divergence of ∑ (log log φ(n))^{ℓ-1}/(φ(n) log φ(n)).
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The central discovery is that the log-primorial sequence $\vartheta_n = \ln([p_n]\#)$ has provably regular gaps: $\vartheta_{n+1}-\vartheta_n = \ln p_{n+1}$, so the relative gap lies in the narrow band $9/(10n) < \mathfrak{g}_n/\vartheta_n < 9/(5n)$, and also $(22/25)\ln p_n/p_n < \mathfrak{g}_n/\vartheta_n < 2\ln p_n/p_n$. From these gap bounds, the paper derives explicit upper and lower bounds on the counting function $\pi_\vartheta(x) = \#\{i : \vartheta_i \le x\}$, then uses them to prove $\vartheta$-analogues: $\vartheta$-Andrica holds with maximal step $\sqrt{\ln 6}-\sqrt{\ln 2}\approx 0.506 < 1$; $\vartheta$-Firoozbakht holds for $n\ge 4$; $\vartheta$-Fourges holds for all $n\ne 2$; $\vartheta$-Nicholson holds for $n\ge 3$; $\vartheta$-Farhadian holds for $n\ge 5$; and $\vartheta$-Legendre, $\vartheta$-Oppermann, and $\vartheta$-Brocard follow from the counting bound $\pi_\vartheta(x)-\pi_\vartheta(y) > (x-y)/(\ln x + 1.059660101)$. The paper stresses that this does not transfer to the ordinary prime conjectures, since the identity $p_n = \exp(\vartheta_n-\vartheta_{n-1})$ does not naively convert $\vartheta$-gap theorems into prime-gap theorems.
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On the paper's own terms, the central claim is that the map from zero-free regions to prime-number-theorem error terms is captured, up to the value of one constant, by the variational quantity $\omega(x)$. Theorem 3.1 shows that for a Beurling zeta function satisfying Axiom A (generalized integers counted by $Ax+O(x^\theta)$) and a zero-density estimate $N(\sigma,T)\ll T^{A(1-\sigma)^B}(\log T)^C$, any zero-free region $\sigma>1-f(\log|t|)$ inside the regular/slow variation class forces the explicit upper bound displayed in the summary; with the currently best Beurling zero-density estimate this becomes the concrete bound of Theorem 1.5. Theorem 1.6 shows the converse construction: for every admissible $f$ with $1/u=o(f(u))$, there is a Beurling number system with $N_P(x)=Ax+O_\varepsilon(x^{1/2+\varepsilon})$, with infinitely many zeros on $\sigma=1-f(\log|t|)$ and none to the right, and with $\psi_P(x)-x=\Omega_\pm(x\exp(-\omega(x)+(1-\delta)f(u_0(x))u_0(x)))$ for every $\delta>0$. The paper also proves a destructive-interference variant showing the parameter $\varepsilon$ in the oscillation theorem cannot be set to zero, and identifies that the classical method for deriving zero-free regions from growth is sharp up to a constant among extended Beurling systems.
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The paper shows that for certain pairs of normalized error terms, such as those associated with ψ(x) and the weighted sum involving Λ(n)/n, the inequality between them holds for all sufficiently large x if and only if the Riemann hypothesis is true. Assuming both the Riemann hypothesis and the linear independence of the positive imaginary parts of the zeta zeros, the authors calculate that the logarithmic density of the set where two specific error terms have the same sign is approximately 0.9865.
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On the paper's own terms, the central claim is Theorem 3: for $g \ge 2$, $L \in \mathbb{N}$ and $2 \le x \le g^L$, the exponential sum $S = \sum_{n \le x} \Lambda(n) e(f_L(n))$ satisfies $S \ll x g^{-\kappa} (\log x)^4$, where $\kappa = \frac{1}{10} \sigma_\xi(\alpha)$ and $\xi = \lfloor \tfrac{1}{4} \log x / \log g \rfloor$. Theorem 2 then gives the quantitative count of reversed primes of length $L$: writing $\pi_L^-(a,q)$ for the number of $p \in [g^{L-1}, g^L)$ with $\mathrm{rev}(p) \equiv a \pmod q$, one has $\pi_L^-(a,q) = \frac{\rho_g(a,q)}{q} \frac{g^L}{\log g^L}(1 + O(1/L)) + O(g^L \exp(-c \sqrt{L}))$ whenever $q \le \exp(c \sqrt{L})$ and $(a,q,g^2-1)=1$ with $g \nmid (a,q)$. The explicit factor $\rho_g(a,q)$ encodes the necessary coprimality conditions, and the error term is effective, depending only on $g$.
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On the paper's own terms, the central discovery is that a recent large-value method for long Dirichlet polynomials survives the passage to polynomials twisted by primitive Dirichlet characters, provided the character sums are controlled by a new bound for affine transformations with GCD twists. For a separated set $W$ of pairs $(t,\chi)$ on which a character-twisted polynomial of length $N$ attains values $\ge V$, Theorem 1.3 gives $|W|\ll_\epsilon N^2V^{-2}+(qT)^{4/3}N^2V^{-4}$ when $(qT)^{3/4}\le N\le (qT)^{5/6}$, and a further four-term bound for larger $N$. Fed into the zero-detection method, this yields $\sum_{\chi\bmod q}N(\sigma,T,\chi)\ll_\epsilon (qT)^{4(1-\sigma)/(1+\sigma)}$, and the combination with the classical bound for $\sigma\le 5/7$ produces the headline estimate $(qT)^{7(1-\sigma)/3+\epsilon}$. The paper also shows the new exponent pays off arithmetically: for fixed prime $p$, the least prime $p(p^n,k)$ in the progression $k\bmod p^n$ is $\ll_{p,\epsilon}(p^n)^{7/3+\epsilon}$, and the least Goldbach number $G(p,k)\equiv k\pmod p$ is $\ll_\epsilon p^{7/6+\epsilon}$.
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The central claim, stated as Eq. (5), is that the twin prime counting function can be written as $\pi_2(x)\sim \frac{2x}{(\ln x)^2}\cdot \frac{\sum_{t=0}^{\infty}(-2)^t f(t;z)}{(\sum_{t=0}^{\infty}(-1)^t f(t;z))^2}$, where $f(t;z)$ is the degree-$t$ elementary symmetric polynomial in the reciprocals of odd primes up to $z$. The derivation expands the numerator product $\prod_{3\le p\le z}(1-2/p)$ as $\sum_{t=0}^{\infty}(-2)^t f(t;z)$ and the denominator product as $\sum_{t=0}^{\infty}(-1)^t f(t;z)$, so the ratio reproduces $D(z)$ term by term. This recasts the sieve correction factor, and ultimately the Hardy-Littlewood constant, as a structured alternating series instead of an opaque product. The paper's own numerical test, with $z=\lfloor x^{1/4}\rfloor$ and terms through $t=4$, is not accurate for larger $x$; the author attributes the systematic overestimation to truncation and presents the framework as a constructive heuristic whose convergence merits further analysis.
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The central claim is Theorem 1.2: for every $k\ge 2$, $\min\{k,3\} \le \mathrm{bs}(k) \le g(k)+2$, where $g(k)$ is the smallest integer $g$ such that for every $m\in[2,k]$ the binary system of alternating-sum equations $\sum_{i=0}^r (-1)^i \binom{r}{i}\alpha_i=0$ for $r=m-g,\ldots,m$ has only the all-zero and all-one solutions. Plugging in the known values $g(2)=0$, $g(k)=1$ for $3\le k\le 13$, and $g(k)\le G(k)$, the paper obtains $\mathrm{bs}(2)=2$, $\mathrm{bs}(k)=3$ for all $k\in[3,13]$, and, from the prime-gap estimate $G(k)=O(k^{0.525})$, the general bound $\mathrm{bs}(k)=O(k^{0.525})$. The lower bound is proved by constructing, for every $k\ge 3$, two moderately dense $k$-uniform hypergraphs with zero relative discrepancy: one is an $(n,k,k-1,\lambda)$-block design and the other contains exactly the $k$-sets that meet each part of a fixed $(k-1)$-partition, so every placement of the two has the same intersection size, namely $pq\binom{n}{k}$.
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On the paper's own terms, the discovery is Theorem 1: with the TG kernel parameters chosen in Sections 4–6, the total approximation error E(x) satisfies |E(x)| < 1/2 for all x ≥ $10^{3}$, so rounding gives π(x) exactly. The authors attribute the power of the construction to a compactly supported, even, smooth kernel Φ_TG(t) that equals $e^{{-t^2}}$ up to a cutoff α, is tapered to zero by a cubic polynomial over a short interval, and is normalized so that its zeroth moment vanishes, forcing F_TG(1)=0 and killing the main term in the explicit formula. Each error source is bounded with explicit constants: the Gaussian tail by (α+Δ)$e^{{-α^2}}$, the omitted high zeros by 0.6C(ln T+1)/$T^{2}$ using an unconditional zero-density estimate, and the trivial-zero terms below $10^{-6}$. The authors conclude that the formula is not merely asymptotic but numerically effective, giving a conservative total error far below 1/2 in their worked example. This is the result they want a fair reader to accept: a rigorous, assumption-free route from zeta zeros to exact values of π(x).
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The central discovery is that the arithmetic obstruction to reversing the digits of primes is mild enough to be handled by averaging. For primes $p$ in a $\lambda$-digit interval, the count of $p$ with $R_\lambda(p) \equiv a \bmod d$ is $\pi_\lambda(t)/d$ plus an error that, summed over all $d \le b^{\xi\lambda}$ with $\gcd(d, b(b^2-1)) = 1$ and all residue classes, is $\ll b^{\lambda - c\sqrt{\lambda}}$. Once this Bombieri-Vinogradov statement is in place, the linear and weighted sieves give Theorem 1.1. In addition the paper obtains an upper bound of the same shape $\ll b^{\lambda}/\lambda^2$ for primes whose reverse is itself prime, and a Siegel-Walfisz-type range $d \le \exp(c\sqrt{\lambda})$ in which the asymptotic holds for individual moduli.
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Stated on the paper's own terms: for primes, the multi-scale sequences $f^{(k)}(n)$ take values in $\{0,\dots,255\}$, and for a sufficiently large interval $[1,m]$ the histograms of the $k$-patterns, for every $k\ge 2$, share the same outline, a shape that remains invariant as $m$ grows and that the paper treats as a characteristic of prime numbers. The count of each histogram bin is approximated by $C_j^{(k)}(m)\,\frac{m}{8^k}\,q_{k-1}(m)^{j}\,(1-q_{k-1}(m))^{8-j}$, where $j$ is the number of ones in the pattern, $q_{k-1}(m)=1-(1-p(m))^{4\cdot 8^{k-2}}$, and $p(m)=2\,\mathrm{li}(m)/m$; the empirical factors $C_j^{(k)}(m)$ are bounded (below 2 for the tested levels, and inside $[1/2,3/2]$ for the first-level factors when $m>16{,}777{,}216$) and absorb the correlations that a strict independence model misses. At the finest scale the admissible patterns are exactly the 14 masks compatible with divisibility by 3, and for Mersenne primes the first-level patterns reduce to $\{0,2,8,10,42,128\}$, with no observed block containing more than one Mersenne prime.
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The paper's central claim is that $S$ is dense in $(0,1)$, made precise by Theorem 3: for every $t \in (0,1)$ and every $\varepsilon > 0$, there exist primes $p > q$ with $\left|\frac{p-q}{p+q} - t\right| < \varepsilon$. The construction proceeds by mapping the target $t$ to the required ratio $r = \frac{1+t}{1-t}$ of the two primes, then searching for primes near $r q$. For targets not handled by this targeted search, a direct search over all prime pairs up to a bound $N_0$ is invoked, with $N_0$ chosen so that the average spacing of the values $\frac{p-q}{p+q}$ is below the tolerance. The paper also states complexity bounds for the search size, namely $N = O\left(\frac{1}{\varepsilon}\log\frac{1}{\varepsilon}\right)$ in the worst case.
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The central claim is Theorem 1.3: for every $0<\theta<1$, the exceptional-set exponent satisfies $$\mu(\$\theta$) \le \inf_{\varepsilon>0} \sup_{0\le\$\sigma$<1,\ A(\$\sigma$)\ge 1/(1-\$\theta$)-\varepsilon} \min(\mu_{2,\$\sigma$}(\$\theta$), \mu_{4,\$\sigma$}(\$\theta$)),$$ where $\mu_{2,\sigma}(\theta)=(1-\theta)(1-\sigma)A(\sigma)+2\sigma-1$ and $\mu_{4,\sigma}(\theta)=(1-\theta)(1-\sigma)A^*(\sigma)+4\sigma-3$. Here $A(\sigma)$ is the least exponent such that $N(\sigma,T)\le T^{A(\sigma)(1-\sigma)+o(1)}$ counts zeroes of the Riemann zeta function in the rectangle $\beta\ge\sigma,\ |\gamma|\le T$, and $A^*(\sigma)$ is the analogous exponent for $N^*(\sigma,T)$, which counts quadruples of such zeroes whose imaginary parts satisfy $|\gamma_1+\gamma_2-\gamma_3-\gamma_4|\le 1$. The minimum of the two expressions corresponds to two alternative routes through the explicit formula: a second-moment bound controlled by ordinary zero counts and a fourth-moment bound controlled by the additive energy of zeroes; the supremum restricts attention to the zero-density bands that are large enough to threaten the prime number theorem. Inserting the best available estimates for $A(\sigma)$ and $A^*(\sigma)$ gives, in particular, $\mu(17/30)\le 7/12$ and $\mu(2/15+\Delta)\le 1-(9/13)\Delta$ for sufficiently small $\Delta>0$. The same framework recovers the classical thresholds for the prime number theorem in all short intervals and in almost all short intervals, and recovers the conditional bounds $\mu(\theta)\le 1-\theta$ under the Riemann hypothesis, $\mu(\theta)\le 1-\theta/2$ under the Lindelöf hypothesis, and $\mu(\theta)\le 1-\theta/12$ under the density hypothesis.
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Theorem 1.1 (Smooth weighted PNT and ZFR): If zeta(sigma + i t) is nonzero for sigma > 1 - eta(log|t|), then Delta(x) is bounded by x exp(-(1-epsilon) omega_eta(x)); conversely, if Delta(x) is bounded by x exp(-(1-epsilon) varpi(x)) and eta satisfies lim eta'(u) = 0, then zeta has no zeros in sigma > 1 - eta(log|t|) for large t. This equivalence is the load-bearing assertion.
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On the paper's own terms, the discovery is Theorem 1: every Catalan number beyond the sixth is divisible by at least one prime of the form $6k-1$. The route goes through the interval $(n+1, 2n]$: because $C_n$ is a quotient of factorials, every prime inside that interval survives into its factorization. If $C_n$ had no $6k-1$ prime factor, all of those interval primes would have to be congruent to $1$ mod $6$, which the author takes to contradict the distribution of primes and the existence of twin primes. From this the paper derives Corollary 4, that $6$ divides $\sigma(C_n)$ for $n>6$, and it independently proves Lemma 2, that $\sigma(6k-1)$ is divisible by $6$ for every positive integer $k$.
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The paper's central claim is that a single very general definition of Ramsey number subsumes the classical numbers $R(z_1,\ldots,z_m)$, the usual generalized graph Ramsey numbers, and several problems not normally seen as Ramseyian. For a Ramsey base $(G,S)$—a hereditary family of graphs together with chosen sets of admissible edge-colorings—and a Ramsey symbol $(X,C)$ of colored target graphs, the Ramsey number $R_{(G,S)}(X,C)$ is the first index $n$ such that every admissible coloring of $G_t$ with $t\ge n$ contains one of the target subgraphs with the prescribed coloring. When the colors form a finite field $\mathbb{F}_q$, the paper defines an indicator polynomial $p[G_i,X,C]$ that vanishes exactly on colorings containing a target subgraph, and Theorem 3 identifies the Ramsey number, in the maximal-base Galois case, as $i_m+1=i_k$, where $i_m$ is the last index with a nonzero indicator and $i_k$ the first index with an identically zero one. The final theorems then encode the Green-Tao theorem, the twin prime conjecture, Zhang's bounded prime gap theorem, and Polignac's conjecture as existence statements for Ramsey numbers of this kind. In the author's framing, this shows that the structure of Ramseyian objects, even when numeric values are out of reach, is tractable.
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The central claim is that the Gauss Circle Primes—values of $C(r)$ that are prime—satisfy $\kappa(n)\sim n/\log n$, matching $\pi(n)$. Tables 2 and 3 show $\kappa(n)$ close to $\pi(n)$ throughout the computed range up to $2\times 10^6$, and Table 4 shows $\pi(n)/\kappa(n)$ closer to 1 than $\pi(n)/(n/\log n)$. Heuristic Result 3.1 states the stronger claim $\kappa(n)\approx\pi(n)$ for all $n$. The mechanism is that each $C(k)$ is odd (indeed $1\bmod 4$), so under the Prime Number Theorem its conditional probability of being prime is about $2/\log C(k)$; using $C(k)\approx\pi k^2$ turns the sum into about $n/\log n$.
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The central claim is Theorem 1.2: for every positive integer $n$, the interval $(n^2,(n+1)^2)$ contains an integer $a$ with $\Omega(a) \le 4$, where $\Omega(a)$ counts prime factors with multiplicity. The proof splits at $N = 1.98\cdot10^{28}$. For smaller square endpoints, a cited computation gives primes in each half of the square interval, and the observation that $4p$ lies between $n^2$ and $(n+1)^2$ when $p$ lies in a suitably rescaled interval lifts the coverage up to the cutoff. For larger $N$, the authors sieve the set $\mathcal{A} = \mathbb{Z} \cap (N, N+2\sqrt{N})$ with $z = X^{1/8}$, $y = X^{1/4}$, where $X = \lfloor N + 2\sqrt{N}\rfloor$, and apply an explicit weighted-sieve inequality to show that the count $r_4(\mathcal{A})$ of elements with at most four prime factors is positive. This establishes the first unconditional result of this form valid for all $n$ rather than only for sufficiently large $n$.
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The paper's central claim is a lower-bound, sieve-theoretic result: for all large $x$, the number of primes $p\le x$ of the form $p=m^2+n^2+1$ whose shift $p+2$ has at most 11 prime factors is $\gg x/(\log x)^{5/2}$. The argument decomposes the sieving by the set of primes congruent to $3$ modulo $4$: the setup forces $p-1$ to have exactly one factor of $2$, so the representation condition reduces to the absence of $3\pmod 4$ prime factors in the odd part. A vector-sieve fundamental lemma supplies lower and upper bounds for two simultaneous $\beta$ sieves of dimensions $1/2$ and $1$; the Buchstab term is handled by a switching principle; and a logarithmic weight function is optimized numerically with the parameters $\theta_2=0.011$, $\theta_1=0.449$, $\theta=0.23$, and $\lambda=0.14$, producing the bound 11.
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The paper's central claim is the Second Fundamental Theorem of the Quanta Prime Sequence (Theorem 25): for any nonzero point $(\alpha,\beta)$ and any natural number $n$, the ratio $\Omega_0(\lfloor n/2\rfloor|\alpha,\beta|n)$ over $(n-1)(n-2)\cdots(n-\lfloor n/2\rfloor)$ equals $\Psi(\alpha,\beta,n)$ and is an integer. From this single identity the author derives new quotient representations for Mersenne numbers, Fermat numbers, Lucas numbers, the Fibonacci-Lucas oscillating sequence, Chebyshev polynomials, and Dickson polynomials, together with the divisibility statement $p_{k+1}\mid \Omega_0(p_k|\alpha,\beta|2p_k)$ for the $k$th prime $p_k$. The paper also presents this as the proof of the previously unproved Theorem 9 from the prior paper [3], recasting Mersenne primality in terms of divisibility of $\Omega$-quotients.
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The central claim is that for every k, after choosing any two residue values modulo each prime up to p_k, at least one integer in [$p_k^{2}$, p_{k+1}^2] lies outside all the chosen classes. The proof rests on the asymptotic P(x) = C_2 / (ln x)^2 with C_2 ≈ 1.07 for the product over odd primes of (1 − 2/p). Substituting x = $p_k^{2}$ and x = p_{k+1}^2 and subtracting produces Δ_k ~ (C_2 h / 2) p_k / (ln p_k)^2 under the fixed-gap assumption p_{k+1} − p_k = h ≥ 2. Because the main term is positive and unbounded, the author concludes that the covered set cannot exhaust the interval, so an uncovered integer must exist. The manuscript also notes that restricting to odd candidates halves the raw estimate and that the same scheme extends to products ∏(1 − c/p).
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On the paper's own terms, the central claim is Theorem 3.1. With $K(x)=\sqrt{\alpha/2\pi}\,e^{-\alpha x^2/2}$ and $I(\omega,\eta)=\int_{\omega-\eta}^{\omega+\eta}K(u-\omega)\,u e^{-u/2}(\pi(e^u)-\mathrm{li}(e^u))\,du$, the theorem asserts that if all zeta zeros with imaginary part up to height $A$ lie on the critical line, and if $5A/(4\omega)\le\alpha\le A^2$ and $0<\eta<\omega/100$, then for $2\pi e\le T\le A$, $$I(\omega,\eta)\ge -1-\sum_{0<|\gamma|\le T}$e^{{i\omega\gamma}}$\left(\frac{1}{\rho}+\frac{1}{\omega\$rho^{2}$}\right)$e^{{-\gamma^2/2\alpha}}$+R,$$ with $|R|\le R_1+\cdots+R_6$. The new $R_6$ term, $\left(1+\frac{22}{A\omega}\right)A\log A\left(\frac{8.283}{A}e^{-\frac{\alpha}{4}\eta^2+\frac{\omega+\eta}{2}}+7.152\,\eta e^{-\frac{A^2}{2\alpha}+\frac{\omega+\eta}{2}}\right)$, improves the older $S_6$ term because its leading exponential has $\alpha/4$ rather than $\alpha/8$ in the exponent after substituting the old lower bound for $\eta$, and because the large factor $A$ cancels in the first exponential; it also makes the lower condition on $\eta$ unnecessary. If the Riemann hypothesis holds in full, $R_6$ and the parameter conditions are omitted.
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The paper's central claim is Theorem 1.2: if some fixed $c_2>0$ has the property that $L(s,\chi)$ never vanishes in $\Re(s)>1-c_2$ for any Dirichlet character $\chi$, then for $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ and base $g$ sufficiently large, almost all even numbers $m \in [X,X+H]^*$ are Goldbach numbers; equivalently, the set of even exceptions has size $o(|[X,X+H]^*|)$. The proof follows the classical circle-method scheme for Goldbach in short intervals: the unit circle is split into Farey arcs, the major arcs give the main term $\mathfrak{S}(-2n) M^*(2n)$ built from Ramanujan sums, and the minor arcs are controlled by combining an upper bound for the weighted prime exponential sums with an $L^1$-average bound for the missing-digit exponential sum. A supporting result, Theorem 1.1, is unconditional: for $X^{3/5+\varepsilon} \ll H \ll X^{1-\varepsilon}$, the average of $d_4(n)$ over $[X,X+H]^*$ is $O((\log X)^7 |[X,X+H]^*|)$. This divisor bound is used to show that the average of a Ramanujan-sum tail in the major-arc analysis is $o(|[X,X+H]^*|)$, which is what makes the 'almost all' conclusion quantitative.
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The main result, Theorem 2.7, states that for each $k \in \mathbb{Z}^+$, as $x \to \infty$, $$\frac{1}{\mathrm{li}_k(x)} \sim \frac{(\log x)^k}{x}\left(1 - \frac{k}{\log x} - \sum_{n\ge 1}\frac{$a_n^{{(k)}}$}{(\log x)^{n+1}}\right),$$ where the coefficients $a_n^{(k)}$ are fixed by the recurrence $$(k-1)!\,$a_n^{{(k)}}$ = (n+k)! - k(n+k-1)! - \sum_{m=0}^{n-2} (n-m+k-2)!\,a_{m+1}^{(k)},$$ with $a_0^{(k)}=1$. The proof starts from a formal power-series identity for the generating function of these coefficients and transfers it into an asymptotic statement via the standard asymptotic expansion of the exponential integral and a change of variables. When $k=1$, the coefficients count indecomposable permutations and the expansion reduces to the known reciprocal prime-counting expansion.