The Fine-Structure Hierarchy of Prime Biases and the Universal Dominance of −1 pmod N
Abstract
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 S̃T(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 − χ0̅(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 \textbf{fine-structure bias} (or \emph{secondary bias}) strictly governed by the special values log L(1, χ). We prove that S̃T(x, χ1,a) := S̃T(x, 1) − S̃T(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.
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BibTeX
@misc{koyama2026finestructure,
title = {{The Fine-Structure Hierarchy of Prime Biases and the Universal Dominance of $-1 \pmod N$}},
author = {Shin-ya Koyama},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.28931},
eprint = {2607.28931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.28931},
}
RIS
TY - UNPB
TI - The Fine-Structure Hierarchy of Prime Biases and the Universal Dominance of −1 pmod N
AU - Shin-ya Koyama
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2607.28931
UR - https://arxiv.org/abs/2607.28931
AB - 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 p^2 ≡ 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 S̃_T(x, a) and adopting the virtual character χ_1,a(x) := 1_{x ≡ 1 pmod N} − 1_{x ≡ a pmod N}, the principal character component χ_0 cancels identically since 1 − χ_0̅(a) = 0. This automatic algebraic elimination erases both the universal logarithmic growth log x and the background noise log L(1, χ^2). Under the Deep Riemann Hypothesis (DRH), we uncover a hitherto undetected \textbf{fine-structure bias} (or \emph{secondary bias}) strictly governed by the special values log L(1, χ). We prove that S̃_T(x, χ_1,a) := S̃_T(x, 1) − S̃_T(x, a) = C_N · log L(1, χ_1,a) + 𝒪((log x)/√x) as x → ∞, where C_N > 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.
ER -