Sharp Local Densities and Signed Weights for Quadratically Weighted Farey Fractions
Year: 2026
Type: preprint
Abstract: For Farey fractions b/q of order Q and the quadratic weight w(q/Q) = (1 − q/Q) 2 , we determine sharp maxima of local masses in arbitrary intervals of length h/Q2 . For the unit weight, assuming h → ∞ and h/Q → 0, the normalized maximum is 8/125 + O(h −1/2 log h + h/Q). For the Gauss weight w(q/Q)φ(q)/q, under the addi tional sufficient condition h/ log log Q → ∞, the normalized maximum is 1/16 + o(1). A Farey-line parametrization supplies the upper bounds for every interval, while explicit spoke constructions and anchors absorbing increasingly many primes give matching lower bounds. For primitive Dirichlet characters, the corresponding normalized constants are 8π 4 /375 and π 4 /48. We also prove absolute domination of associated signed Farey weights by the Gauss weight and show that the raw, positive-part, and absolute-value signed sums all retain the sharp worst-window constant 1/16. A Mellin–Landau argument proves infin itely many sign changes in the real smoothing parameter. Finally, a concise fixed-support variational test certifies three global minimizers by strict convexity, exact Euler–Lagrange equations, and interval arithmetic; this test does not reconstruct the family-average error chain in the motivating zeta-zero application.
Author: Ye, Di
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225599164