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
Year: 2026
Type: preprint
Abstract: Manuscript, code and data for the second part of "Sums of two squares with few prime factors", version 3 (1 November 2026). Part I is doi:10.5281/zenodo.23826176, Part III is doi:10.5281/zenodo.23826860. Let m, m′ be coprime odd integers, Θ_{m,m′}(z) = θ²(mz) θ²(m′z), G_{m,m′} its cuspidal part and L = 4mm′, so that a_G(N) = T(m,m′;N) − M(m,m′;N) is the error term in the number of representations of N by m(x² + y²) + m′(z² + w²). Part I proved ‖G_{m,m′}‖² ≪ (mm′)^ε for the Petersson norm and deduced the level of distribution 1/2 for the weighted sequence r(n) r(N − n) and the bounds Ω(n(N − n)) ≤ 8, max(Ω(n), Ω(N − n)) ≤ 5 for every large N ≡ 2 (mod 4). (1) Mean square. For every G ∈ S₂(Γ₀(L)), Σ_{N≤Y} |a_G(N)|² ≤ 4πe² Y (1 + 54Y/L) ‖G‖² (a covering argument); for G_{m,m′} the root mean square of (T − M)/√N is ≪ (mm′)^{−1/2+ε}, which gives the level of distribution 1 − ε for the weighted sequence for all but O(X^{1−δ}) integers N ≤ X, and for those N the bounds 5 and 4 (as an arithmetic statement about almost all N this is superseded by the circle method). The uniform bound of Part I is reformulated in terms of Poincaré series, the individual bound |T − M| ≪ N^{1/2+ε}(mm′)^{−1/2+ε} suggested by the data is stated, and any power saving in mm′ for individual N is shown to raise the level of distribution for every N beyond 1/2. (2) The size of the norm. The norm depends only on mm′ (an Atkin–Lehner involution maps G_{m,m′} to G_{1,mm′}); the lower bound ‖G_{m,m′}‖² ≥ (1/π)(log mm′)² − O(log(2mm′)(log log 3mm′)^C) comes from the regions of the fundamental domain above the Atkin–Lehner cusps, the matching upper bound ‖G_{m,m′}‖² ≪ (log 3mm′)²(log log 3mm′)^C from the local densities, so the norm is of the exact order (log mm′)² when mm′ has boundedly many prime factors and the uniform bound for T − M holds with (log 3mm′)(log log 3mm′)^C in place of (mm′)^ε. Evaluating the Parseval sums over the cusps at the heights 1 and √3/2 gives the asymptotic formula ‖G_{m,m′}‖² = (1/π) Σ_{δ|mm′, δ²<mm′} δ^{−1}(log(mm′/δ²))² + O(log(2mm′)(log log 3mm′)^C) for square-free mm′, in particular ‖G_{1,p}‖² = (1/π)(log p)² + O(log p). (3) Rankin–Selberg. By an explicit unfolding, ⟨Θ_{m,m′}, f(d·)⟩ = ±(4/π²) Π_{p | mm′/M₁}(1 + χ_{−4}(p)) d^{−1} L(1,f) L(1,f⊗χ_{−4}) for every newform f of level 2^e M₁ | 4mm′ and every d, with no other local factors; the norm bounds are therefore bounds for the second moment of L(1/2,f) L(1/2,f⊗χ_{−4}) over the family, and the asymptotic formula is an asymptotic formula for this second moment. Every newform of level 4n (n odd) has η₄ = −1, so when χ_{−4}(mm′) = −1 no newform of level 4mm′ occurs in G_{m,m′} and L(1,φ) L(1,φ⊗χ_{−4}) = 0 for all these newforms. (4) Numerics. The regions above the cusps account for 93–98% of the norm; there is an exact check at level 20; the Rankin–Selberg formula is verified for newforms at the levels 20, 52, 116, 148 and for oldforms inside them; the supremum of |T − M|/√N exceeds the root mean square by a bounded factor at every level computed. (5) The sieve. Replacing the vector sieve of Part I by the two-dimensional sieve of Diamond, Halberstam and Richert gives Ω(n(N − n)) ≤ 7 with max(Ω(n), Ω(N − n)) ≤ 5 for every large N ≡ 2 (mod 4) from the same level 1/2; the two inequalities for F₂, f₂ that this needs are certified in interval arithmetic. The symmetric remainder of the two-dimensional sieve is the projection onto the joint (+1)-eigenspace of the Atkin–Lehner involutions W_p, p | d; Selberg-type mollifiers below the sifting limit are identically 1 on the sifted integers. Section 8 makes the Kloosterman structure of the remainder beyond level 1/2 exact: the Fourier transform of the circle x² + y² ≡ a modulo any odd c is a classical Kloosterman sum, and the leading term of a_G(N) − a_G(N/2) is a sum of S(inv(16m)·N, |h|²; m′k) over the moduli c ≡ 0 (mod m′), the inverse of 16m taken modulo the modulus. Part III takes up the dispersion form, the sieve barriers and the short-interval mean squares. Version 3 corrects the Kloosterman sums of Section 8.1 and the introduction: they are S(inv(16m)·N, |h|²; m′k), not S(16N, m|h|²; m′k) as in the earlier versions (a Kloosterman sum depends only on the product of its two arguments); the Kuznetsov assessment of Remark 8.3(ii) is withdrawn. Lemma 8.1 now covers all odd moduli, Proposition 8.2 is stated as the leading term, and the bibliography records Part III. Part IV of the series discusses the correction and audits its consequences for Parts II and III. The deposit contains the paper (PDF), the working notes, the Python scripts and their outputs. Source repository: https://github.com/Ruqing1963/two-squares-mean-square (tag v1.0 = commit 329bdb7).
Author: Chen, Ruqing
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225016232