Rational directions as point masses: a window law and a first moment of angular Hecke L-functions for Q(√5)
Year: 2026
Type: preprint
Abstract: In the integer lattice ℤ², a thin sector around a line of rational slope contains a whole line of lattice points through the origin, and these points produce a term that the area of the sector does not see (Waxman and Yesha, Monatsh. Math. 204 (2024)). This paper studies the same phenomenon for the ring of integers ℤ[φ] of the real quadratic field ℚ(√5), with three differences: the weight is N(I)^(−s), so the objects are Dirichlet series; the natural "direction" of an ideal I = (α) is its position v(I) = log|α/α′|/(2 log φ) mod 1, which is invariant under the units; and the Fourier modes of the position are the angular Hecke characters λ_k. A statement about windows in v is therefore at the same time a statement about the L-functions L(s, λ_k). MAIN RESULTS (all proved, s real with 1/2 < s < 1 where relevant; T_s = ζ(2s)(1 + 5^(−s))) 1. Point masses (Propositions 3.1 and 3.3). The only rational position is 0, taken exactly by the ideals (m) and (m√5). Every ideal factors uniquely as (m)(√5)^e·J with J primitive and has the position of J; distinct primitive ideals have distinct positions. Hence Σ_I N(I)^(−s) δ_v(I) = T_s · Σ_J N(J)^(−s) δ_v(J): the weighted position measure is a sum of point masses, and the unit ideal gives the mass T_s at v = 0. 2. Uniform counting (Lemmas 4.1–4.3). One lattice-point bound, |#{I : N(I) ≤ Y, v(I) ∈ J} − |J|·ϱ·Y| ≤ κ√Y uniformly in the interval J (ϱ = 2 log φ/√5), gives a uniform bound for the window discrepancy and the bound |L_X(s, λ_k)| ≤ κ(1 + 2π|k|)(X^(1/2−s) + s/(s − 1/2)) for the partial sums of the L-series, which converge for Re s > 1/2. 3. Window law, exact form (Theorem 5.1). The regularised window mass C_s(w) = lim_X [Σ_{N(I) ≤ X, |v(I)| < w} N(I)^(−s) − 2wϱ X^(1−s)/(1−s)] exists for 0 < w ≤ 1/2 and equals C_s(w) = T_s + ζ(2s−1)/(2s−1) · 4^s (5^(1/2−s) + 5^(−1/2)) · t^(2s−1) · ₂F₁(s, s−1/2; s+1/2; t²) + ρ_s(w), with t = tanh(w log φ) and |ρ_s(w)| ≤ 2T_s (4t²/(1−t²))^s. The value ζ(2s) (inside T_s) comes from the lattice points on the rational line α ∈ ℤ ∪ √5ℤ; the value ζ(2s−1) comes from the infinitely many rows of lattice points parallel to it, through the regularised row sum Σ_y y^(1−2s). Both families of rows lead to the same hypergeometric function. 4. Asymptotic form (Corollary 5.2). T_s − C_s(w) = c_s·w^(2s−1) + O(w^(2s)) with c_s = −ζ(2s−1)/(2s−1) · 4^s (5^(1/2−s) + 5^(−1/2)) (log φ)^(2s−1) > 0. At s = 3/4: T = ζ(3/2)(1 + 5^(−3/4)) = 3.3936578…, c = 4|ζ(1/2)|·√(2 log φ)·(5^(−1/4) + 5^(−1/2)) = 6.395105…, so T − C(w) ∼ c√w. 5. Hecke side (Theorem 6.1 and Corollary 6.2). For every even real Schwartz function Φ, Σ_k Φ(k/K)(L(s, λ_k) − T_s) = −((2s−1)c_s/2) · K^(2−2s) · ∫ Φ̂(v)|v|^(2s−2) dv + O(K^(1−2s)), with an explicit error constant. The secondary term K^(2−2s) is the Fourier image of the singularity w^(2s−1) of the window mass at the point mass v = 0. In particular the smoothed mean of L(s, λ_k) over |k| ≲ K is T_s + O(K^(1−2s)). At s = 3/4 the main term is −(c/4)√K ∫ Φ(x)|x|^(−1/2) dx. 6. A second field (Section 9, proved and checked numerically). For ℚ(√2), with the unit ε = 1 + √2 of norm −1, the same proof gives T_s = ζ(2s)(1 + 2^(−s)) and c_s = −2ζ(2s−1)/(2s−1) · (2^(1/2−s) + 2^(−1/2)) (log ε)^(2s−1), e.g. c_{3/4} = 8.489266. Both fields fit one formula in terms of the half transverse steps of the rows parallel to the two rational lines; this common shape is stated as a conjecture for real quadratic fields of class number one with a unit of norm −1. METHOD The window is split into three families: the point mass (α ∈ ℤ or α ∈ √5ℤ), the lattice points of positive norm in rows parallel to the line ℤ, and those of negative norm in rows parallel to √5ℤ. Each row is compared with half its integral by a monotonicity lemma whose error is the value at the start of the row; the row integrals have a closed hypergeometric form; the sum over the rows is regularised by Abel summation, which produces ζ(2s−1); and the smooth part is computed in hyperbolic coordinates x = 2r cosh θ, √5y = 2r sinh θ. The Hecke side follows by Stieltjes integration of the window mass against the smoothing kernel Σ_k Φ(k/K)e(ku), with dominated convergence justified by the uniform bounds of Section 4. The only external input is the classical lattice-point theorem for homogeneously expanding regions with Lipschitz boundary (Lang, Algebraic Number Theory, Ch. VI §2). NUMERICAL VERIFICATION All 4 304 070 ideals of ℤ[φ] and all 6 232 266 ideals of ℤ[√2] of norm at most 10⁷ were enumerated. Selected results:• ℚ(√5): (T − C(w))/√w = 6.395517 at w = 0.001 against c = 6.395105; (T_s − C_s(w))/w^(2s−1) at w = 10⁻⁴ is 9.45767 and 10.46346 against c_s = 9.45763 (s = 0.6) and 10.46341 (s = 0.9); the exact form agrees with the direct sum with norm cutoff 10⁷ to within 5·10⁻⁴.• ℚ(√5), first moment with Gaussian weight, K = 5 … 300: ratio of the sum to the main term between 0.999931 and 1.000021 at s = 3/4 and between 0.999949 and 1.000002 at s = 0.9.• ℚ(√2): measured constants 11.737101, 8.489227, 14.690488 against the predicted 11.737136, 8.489266, 14.690529 (s = 0.6, 0.75, 0.9); first-moment ratios 0.99973–0.99991 (s = 3/4) and 0.99986–0.99997 (s = 0.9). Two naive transfers of the ℚ(√5) constant miss the measured value by 13 % and 41 %. STATEMENTS THAT FAIL (Section 8, negative) • A pointwise law L(3/4, λ_k) ≈ T − (c/4)k^(−1/2) is false; individual values carry the Fourier coefficients of all other point masses, and only the smoothed sum isolates the mass at 0.• The window does not give a fast route to ζ(3/2): at X = 10⁷ the window constant misses T by 0.2019, while the point mass alone misses it by only 0.0933.• C_s(w) has no expansion in integer powers of w; the leading singular term is w^(2s−1) and the next is w^(2s). STATUS OF THE RESULTS Every result in the paper carries one label: proved; proved, cited (classical statements with reference); numerical; conjecture; open; negative. Open questions: the window law around the other point masses v(J), J ≠ 𝒪; fields with a unit of norm +1; a derivation from the constant term of an Eisenstein series (stated as a conjecture: the pair ζ(2s), ζ(2s−1) is the same as in the constant term for SL₂(ℤ)); the Gaussian analogue for ℤ[i]. The window law and a first moment of the angular L-functions of a real quadratic field with secondary term K^(2−2s) were not found in the literature searched (arXiv and the journals cited). FILES • RD_Rational_Directions_EN.pdf – the paper (13 pages).• RD_Rational_Directions_EN.tex – LaTeX source (pdfLaTeX, compiles on Overleaf, no figure files).• RD_companion_code.zip – Python scripts rd_checks.py (ℚ(√5), 24 checks) and rd_sqrt2_checks.py (ℚ(√2), 17 checks), driver run_all_checks.py, logs, result files, README and SHA256SUMS. Requirements: Python ≥ 3.10, numpy, mpmath. Run time about one minute on two cores. License: CC BY 4.0.
Author: Krause, Thomas
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222453833
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