2026/07/31 by Shin-ya Koyama
Mathematics · #math.NT #msc:11N13 #msc:11M26 #msc:11M41 #msc:11N69
10 pages, 4 tables. This is a revised and expanded version of the author's previous manuscript "A Hidden Hierarchy of Chebyshev's Bias and the Dominance of -1 (mod N)" (submitted April 26, 2026)
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We investigate the deterministic hierarchy of prime distribution biases in arithmetic progressions modulo N using a regularized spectral approach. Classical studies on Chebyshev's bias attribute prime races primarily to the accumulation of prime squares p2 ≡ 1 \pmod N, which creates a systematic deficit in quadratic residue classes. However, this classical mechanism fails to explain or distinguish any bias among residue classes sharing identical quadratic residue status (e.g., 3, 5, 7 \pmod 8). To overcome the long-standing analytical obstacles of jump discontinuities and non-convergent boundary fluctuations inherent in classical Perron-type step-function truncations, we introduce a smooth C^∞ Gaussian mollifier into Weil's explicit formula for Dirichlet L-functions. By defining the spectrally normalized individual mollified sums \widetilde ST(x, a) and adopting the virtual character χ1,a(x) := 1_\x ≡ 1 \pmod N\ - 1_\x ≡ a \pmod N\, the principal character component χ0 cancels identically since 1 - χ0(a) = 0. This automatic algebraic elimination erases both the universal logarithmic growth log x and the background noise log L(1, χ2). Under the Deep Riemann Hypothesis (DRH), we uncover a hitherto undetected fine-structure bias (or secondary bias) strictly governed by the special values log L(1, χ). We prove that \widetilde ST(x, χ1,a) := \widetilde ST(x, 1) - \widetilde ST(x, a) = CN ⋅ log L(1, χ1,a) + O((log x)/√ x) as x → ∞, where CN > 0 depends solely on N. Consequently, we establish a deterministic multi-way ranking (such as 7 > 3 > 5 > 1 \pmod 8) that completely transcends the classical quadratic residue framework.