COLLOQUIUM Shin-ya Koyama (小山 信也氏) "Hidden Hierarchy of Prime Biases"

par Prof. Shin-ya Koyama (Toyo University)

jeudi 5 nov. 2026, 16:00 → 17:00 Europe/Paris

René Baire (IMB)

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.