Hidden Hierarchy of Prime Biases

Chebyshev’s bias traditionally describes the phenomenon where primes congruent to 3 (mod 4) outnumber those congruent to 1 (mod 4). While classical results quantified primary biases between quadratic residues and non-residues, fine-structure biases within the same quadratic family remained unformulated.

In this talk, we present a new asymptotic framework that uncovers a hidden hierarchy of fine-structure prime biases among all residue classes modulo $N$. Using spectrally normalized mollified prime power sums weighted over nontrivial zeros, we prove under the Deep Riemann Hypothesis (DRH) that these fine-structure biases are rigorously governed by values of virtual character $L$-functions at $s=1$. We provide a theoretical explanation for the universal dominance of $-1 \pmod N$, revealing a structural dichotomy: odd characters generate systematic biases, whereas even characters are suppressed into noise due to cancellations with trivial zeros. Finally, we address numerical anomalies such as $N=19$ caused by extremely low nontrivial zeros.