Powered by OpenAIRE graph
Found an issue? Give us feedback
image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ ZENODOarrow_drop_down
image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
ZENODO
Review . 2026
License: CC BY
Data sources: ZENODO
ZENODO
Review . 2026
License: CC BY
Data sources: Datacite
ZENODO
Review . 2026
License: CC BY
Data sources: Datacite
versions View all 2 versions
addClaim

Divisor-Filtered Farey Chambers and Localized Discrepancy

Authors: Huckstead, Jeffery;

Divisor-Filtered Farey Chambers and Localized Discrepancy

Abstract

For each integer n >= 1, the elementary chamber grida(n,k) = n - 1 + k/n, for 1 <= k <= n,becomes, after translation to the unit interval and reduction to lowest terms, a divisor-filtered family of reduced fractions. A reduced denominator q appears in chamber n if and only if q divides n, and the number of entries of reduced denominator q is Euler's totient phi(q). This note studies the discrepancy of such reduced fractions when the allowed denominators are restricted to a divisor window D ⊆ Div(n). For 0 <= x <= 1, define A_{n,D}(x) = sum_{q in D} #{1 <= a <= qx : gcd(a,q)=1} and Phi_D = sum_{q in D} phi(q), with discrepancy Delta_{n,D}(x) = A_{n,D}(x) - x Phi_D. We derive a Möbius-inversion formula for Delta_{n,D}, prove the uniform bound sup_{0 <= x <= 1} |Delta_{n,D}(x)| <= sum_{q in D} 2^{omega(q)}, and identify several instructive special cases. The full divisor window D = Div(n) collapses exactly to the regular grid, giving A_{n,Div(n)}(x) = floor(nx). The exact shell D = {n} recovers the classical reduced-residue discrepancy, while a prime-power shell D = {p^a} has the sharper exact supremum 1 - 1/p. These results are finite, elementary, and local. They are not global Farey-discrepancy estimates and carry no implication for the Riemann hypothesis.

Related Organizations
  • BIP!
    Impact byBIP!
    selected citations
    These citations are derived from selected sources.
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    0
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Average
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
Powered by OpenAIRE graph
Found an issue? Give us feedback
selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average
Green