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SUMMARY:COLLOQUIUM Shin-ya Koyama  (小山 信也氏)　"Hidden Hierarchy 
 of Prime Biases"
DTSTART:20261105T150000Z
DTEND:20261105T160000Z
DTSTAMP:20261015T123300Z
UID:indico-event-17347@indico.math.cnrs.fr
DESCRIPTION:Speakers: Shin-ya Koyama (Toyo University)\n\nChebyshev’s bi
 as traditionally describes the phenomenon where primescongruent to 3 (mod 
 4) outnumber those congruent to 1 (mod 4). Whileclassical results quantifi
 ed primary biases between quadratic residuesand non-residues\, fine-struct
 ure biases within the same quadraticfamily remained unformulated.\nIn this
  talk\, we present a new asymptotic framework that uncovers ahidden hierar
 chy of fine-structure prime biases among all residueclasses modulo $N$. Us
 ing spectrally normalized mollified prime powersums weighted over nontrivi
 al zeros\, we prove under the Deep RiemannHypothesis (DRH) that these fine
 -structure biases are rigorouslygoverned by values of virtual character $L
 $-functions at $s=1$. Weprovide a theoretical explanation for the universa
 l dominance of $-1\\pmod N$\, revealing a structural dichotomy: odd charac
 ters generatesystematic biases\, whereas even characters are suppressed in
 to noisedue to cancellations with trivial zeros. Finally\, we address nume
 ricalanomalies such as $N=19$ caused by extremely low nontrivial zeros.\n\
 nhttps://indico.math.cnrs.fr/event/17347/
LOCATION:René Baire (IMB)
URL:https://indico.math.cnrs.fr/event/17347/
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