Binary Goldbach research: circle-method verification, M3C sieve framework, signed Möbius correlations, Lean 4 proofs, and open parity obstructions. No proof of Goldbach claimed.
Bateman–Horn census of prime constellations of Q(n)=n^q−(n−1)^q: 2,393 Q47 quadruplets below 8×10^11 (BPSW-verified), seven-exponent recount, ECPP-certified 1000-digit quadruplets, and a 200,000-digit Q47 PRP. Code, data, certificates, paper v2.
Part IV: the scalloped Parseval formula for the Petersson norm of θ²(mz)θ²(m'z), the closed form of the second coefficient B, the Heegner discriminants of class number one (D = 3, 7, 11, 19, 43, 67, 163, 8), and Eisenstein dispersion. Paper, scripts, results, figures.
Primes p ≤ N with N − p an almost prime sum of two squares (Ω ≤ 9): paper, semi-linear vector sieve with Richert weights and switching, and an interval-arithmetic certification script.
Manuscript, certified computations and data for "Recovering the level of distribution x^{1/2} for the orders of y² = x³ − x modulo p": the Iwaniec–Jiménez Urroz P₂ theorem holds as stated, and Xie's prime-power bounds improve to P₃ and ⌊2.05(ℓ−1)+1.2⌋.
Part II: the mean square of the cuspidal error of θ²(mz)θ²(m'z), the exact order (log mm')² of the Petersson norm, level of distribution 1−ε for almost all N, the Rankin–Selberg formula for the newform components, and the unconditional (7,5): Ω(n(N−n)) ≤ 7, max ≤ 5, n and N−n sums of two squares. Paper (v3), scripts, results.
Harman sieve and Lean-verified certificate for the \(n^{1.323}\) bound on the greatest prime factor of \(n^2+1\).
The dispersion obstacle, sieve barriers at the level 1/2, and short-interval mean squares. The remainder as a trilinear form in Kloosterman sums with a fixed argument, the Atkin–Lehner projection of the symmetric remainder, optimality of Richert's weight (the gap to (7,4) is structural), level Δ−ε in intervals (X, X+X^Δ], spectral decomposition.
Sums of two squares with few prime factors via theta series of quaternary forms. The sieve for r(n)r(N−n) through the cusp form of θ²(mz)θ²(m'z): norm bound ‖G‖² ≪ (mm')^ε, level of distribution 1/2, and Ω(n(N−n)) ≤ 8, max ≤ 5 for every large N ≡ 2 (mod 4). Paper (v3), scripts, certified sieve constants.
A very fast Prime Sieve
Prime quintuplets and the sextuplet boundary for Q(n)=n^47-(n-1)^47 (Titan Project, Part III). Exact Bateman-Horn singular series via Dirichlet L-functions mod 47; parameter-free prediction of 7 quintuplets / 0 sextuplets up to 2e11; first sextuplet expected near N*~1e13. Paper, data, scripts. v2 DOI 10.5281/zenodo.23178170
N = n₁ + n₂ with n₁, n₂ sums of two squares (or values of any class number one form) and Ω(n₁n₂) ≤ 11, max Ω(nᵢ) ≤ 8: paper, linear vector sieve with switching and Richert weights, interval-arithmetic certification.
Finite local sieves for shifted even squares, with exact CRT densities and prime-square extensions.
To associate your repository with the sieve-methods topic, visit your repo's landing page and select "manage topics."