Sums of two squares with few prime factors, II: the mean square of the cuspidal error of θ²(mz)θ²(m′z), almost all N, and the size of the Petersson norm

Manuscript, code and data for the second part of "Sums of two squares with few prime factors", version 1.0 (October 2026). Part I is doi:10.5281/zenodo.23826176. Contents. (1) A mean-square theorem for the Fourier coefficients of a cusp form of level L, Σn≤Y |a(n)|² ≤ 4πe² Y (1 + 54Y/L) ‖G‖², proved by a covering argument; with the norm bound of Part I it gives the level of distribution 1 − ε for r(n) r(N − n) for almost all N ≡ 2 (mod 4), hence Ω(n(N − n)) ≤ 5 and max(Ω(n), Ω(N − n)) ≤ 4 for almost all N. (2) The size of the Petersson norm of the cuspidal part Gm,m′ of θ²(mz)θ²(m′z): ‖Gm,m′‖ = ‖G1,mm′‖, the two-sided bound (1/π)(log mm′)² ≤ ‖G‖² ≪ (log mm′)² (log log mm′)^C, and the asymptotic formula ‖Gm,m′‖² = (1/π) Σδ|mm′, δ²<mm′ δ⁻¹ (log(mm′/δ²))² + O(log mm′ (log log mm′)^C) for square-free mm′, obtained from the Parseval sums over all cusps at the heights 1 and √3/2. (3) An explicit Rankin–Selberg formula, ⟨Θ1,n, φ⟩ = (4/π²) χ₋₄(n) η_n(φ) L(1,φ) L(1,φ⊗χ₋₄) for newforms φ of level 4n and the corresponding formula for oldforms, with no further local factors; the second moment of L(1,f)L(1,f⊗χ₋₄) over the newforms of levels p, 2p, 4p is (π³/16)(log p)² + O(log p); every newform of level 4n (n odd) has Atkin–Lehner eigenvalue η_4 = −1, so that for χ₋₄(n) = −1 the form G1,n has no newform component and L(1,φ)L(1,φ⊗χ₋₄) = 0 for all newforms of level 4n. (4) The two-dimensional sieve of Diamond–Halberstam–Richert applied to n(N − n): Ω(n(N − n)) ≤ 7 and max(Ω(n), Ω(N − n)) ≤ 5 for every large N ≡ 2 (mod 4), with the sieve functions certified in interval arithmetic from the Booker–Browning constants; the remainder at the modulus d is the coefficient of the projection of the cusp form onto the joint (+1)-eigenspace of the Atkin–Lehner involutions W_p, p | d; the dimension of the weighted problem and mollified weights; an LP showing Richert's linear weight optimal. (5) The remainder sum beyond the level 1/2 written exactly as a trilinear form in classical Kloosterman sums S(16N, m|h|², m′k), with numerical evidence of square-root cancellation. The archive contains the manuscript (paper/two_squares_mean_square.pdf, 47 pages), the LaTeX source, the working notes (notes/), the Python scripts (certification with directed rounding, exact Siegel main terms, Rankin–Selberg tests, sieve-function solvers, numerical scans) and their outputs.