A prime-indexed monotone selection of the Riemann zeros: Real-positive rays of ζ, channel saturation, and Gram's law

{"This":[0,89],"paper":[1,256],"introduces":[2],"a":[3,17,113,158,192,200,245],"new":[4],"structural":[5],"labelling":[6],"for":[7,167],"the":[8,11,33,36,46,50,76,92,97,103,109,168,177,181,187,196,219,231,241,252,255,261,268,272],"zeros":[9,182,198],"of":[10,69,75,96,112,195,203,208,226,271],"Riemann":[12,197],"zeta":[13,51,98],"function":[14,52,99,107],"by":[15,87],"attaching":[16],"canonical":[18],"analytic":[19],"arc":[20],"(a":[21],"\\"ray\\")":[22],"to":[23,35,106,123,216,251],"every":[24],"zero":[25],"$\\\\rho":[26],"=":[27,41,127,135],"1/2":[28],"+":[29],"i\\\\gamma$.":[30],"Defined":[31],"as":[32,108,283],"solution":[34],"initial":[37,73],"value":[38],"problem":[39],"$ds/dv":[40],"1/\\\\zeta'(s)$,":[42],"these":[43],"arcs":[44],"represent":[45],"trajectories":[47],"along":[48,129],"which":[49],"takes":[53],"exactly":[54,153],"real,":[55],"positive":[56,224],"values.":[57],"The":[58,72,120,144,174,211,223],"manuscript":[59],"establishes":[60],"four":[61],"principal":[62],"facts":[63],"regarding":[64,280],"this":[65,227],"geometric":[66,115,247],"framework:":[67],"Unification":[68],"Zeta":[70],"Statistics:":[71],"direction":[74],"ray":[77],"is":[78,85,151,183],"$-\\\\pi":[79],"S(\\\\gamma)$,":[80],"and":[81,102,180,266],"its":[82,281],"radial":[83],"scale":[84],"determined":[86],"$1/\\\\vert{}\\\\zeta'(\\\\rho)\\\\vert{}$.":[88],"formulation":[90],"allows":[91],"two":[93],"main":[94],"statistics":[95],"(the":[100],"argument":[101],"derivative":[104],"modulus)":[105],"polar":[110],"coordinates":[111],"single":[114],"object.":[116],"Exact":[117],"Channel":[118],"Saturation:":[119],"rays":[121],"escape":[122],"$\\\\text{Re":[124],"}":[125],"s":[126],"+\\\\infty$":[128],"horizontal":[130],"channels":[131,179,265],"located":[132],"at":[133],"$t":[134],"k\\\\pi/\\\\log":[136],"2$":[137],"(with":[138],"$k$":[139],"being":[140],"an":[141,276],"odd":[142],"integer).":[143],"study":[145],"demonstrates":[146],"that":[147,163],"each":[148],"available":[149],"channel":[150],"occupied":[152,264],"once":[154],"without":[155],"exception,":[156],"yielding":[157],"precise,":[159],"closed":[160],"count":[161],"formula":[162],"has":[164],"been":[165],"verified":[166],"first":[169,262],"200":[170],"zeros.":[171],"Monotone":[172],"Subsequence:":[173],"mapping":[175],"between":[176],"asymptotic":[178],"strictly":[184],"order-preserving.":[185],"Consequently,":[186],"prime":[188],"2":[189],"naturally":[190],"identifies":[191],"monotone":[193],"subsequence":[194],"with":[199,275],"relative":[201],"density":[202],"$\\\\log":[204],"2/\\\\log(T/2\\\\pi)$.":[205],"Geometric":[206],"Origin":[207],"Gram’s":[209],"Law:":[210],"Gram":[212,233],"points":[213,234],"are":[214],"shown":[215],"lie":[217],"on":[218],"same":[220],"curve":[221],"network.":[222],"branch":[225],"network":[228],"passes":[229],"through":[230],"exact":[232],"where":[235],"Gram's":[236],"law":[237,243],"holds":[238],"true,":[239],"placing":[240],"empirical":[242],"within":[244],"precise":[246],"context.":[248],"In":[249],"addition":[250],"theoretical":[253],"framework,":[254],"includes":[257],"numerical":[258],"data":[259],"tracing":[260],"43":[263],"analyzes":[267],"displacement":[269],"statistic":[270],"pairing,":[273],"concluding":[274],"open":[277],"quantitative":[278],"question":[279],"boundedness":[282],"$T":[284],"\\\\to":[285],"\\\\infty$.":[286]}

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-14
DOI
https://doi.org/10.5281/zenodo.22750066
Primary Topic
Random Matrices and Applications
Type
preprint
Controls
|||
ALL TIME
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
preprint

A prime-indexed monotone selection of the Riemann zeros: Real-positive rays of ζ, channel saturation, and Gram's law

Jorge Vicente Romero
Zenodo (CERN European Organization for Nuclear Research)
Random Matrices and Applications
preprint

A prime-indexed monotone selection of the Riemann zeros: Real-positive rays of ζ, channel saturation, and Gram's law

Jorge Vicente Romero
preprint en

Abstract

This paper introduces a new structural labelling for the zeros of the Riemann zeta function by attaching a canonical analytic arc (a "ray") to every zero $\rho = 1/2 + i\gamma$. Defined as the solution to the initial value problem $ds/dv = 1/\zeta'(s)$, these arcs represent the trajectories along which the zeta function takes exactly real, positive values. The manuscript establishes four principal facts regarding this geometric framework: Unification of Zeta Statistics: The initial direction of the ray is $-\pi S(\gamma)$, and its radial scale is determined by $1/\vert{}\zeta'(\rho)\vert{}$. This formulation allows the two main statistics of the zeta function (the argument and the derivative modulus) to function as the polar coordinates of a single geometric object. Exact Channel Saturation: The rays escape to $\text{Re } s = +\infty$ along horizontal channels located at $t = k\pi/\log 2$ (with $k$ being an odd integer). The study demonstrates that each available channel is occupied exactly once without exception, yielding a precise, closed count formula that has been verified for the first 200 zeros. Monotone Subsequence: The mapping between the asymptotic channels and the zeros is strictly order-preserving. Consequently, the prime 2 naturally identifies a monotone subsequence of the Riemann zeros with a relative density of $\log 2/\log(T/2\pi)$. Geometric Origin of Gram’s Law: The Gram points are shown to lie on the same curve network. The positive branch of this network passes through the exact Gram points where Gram's law holds true, placing the empirical law within a precise geometric context. In addition to the theoretical framework, the paper includes numerical data tracing the first 43 occupied channels and analyzes the displacement statistic of the pairing, concluding with an open quantitative question regarding its boundedness as $T \to \infty$.

Zenodo (CERN European Organization for Nuclear Research)
Peace, Justice and strong institutions
Random Matrices and Applications
AI Navigator

Ask Laika to Summarize, Analyze, and Connect papers live on the map.

Summarize Papers & Methodologies

Extract key findings, datasets, and comparative methods across publications.

Benchmark Rankings & Visual Analytics

Rank top research institutions, authors, funders, topics, and journals by Field-Weighted Citation Impact (FWCI) and paper volume with instant charts.

Connect Distant Disciplines

Bridge topological clusters on the map to find hidden collaborative intersections.