almost-primes

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.

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.

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.

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.

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.