PreprintCC BY 4.0arXiv
Weighted Generalizations of Zagier's Phenomenon
Abstract
We study sums associated with the action of PGL2(ℤ) on continuous piecewise polynomial functions with exactly two real roots, both irrational. We form weighted sums of the positive parts of their normalized transforms, using nonnegative weights compatible with translation, reflection, and inversion. Under suitable continuity, finiteness, and convergence assumptions, we prove that these sums are well defined, bounded, $1$-periodic, and continuous on ℝ, and satisfy a reciprocal functional equation. We also extend the construction to finite families of distinct function orbits. Our framework recovers Zagier's constancy result and includes the full family of quadratic sums for which Bengoechea proved convergence.
Related
- Tree Lattices and the Neumann-Zagier conjecture
- Value-distribution of Euler--Zagier type multiple zeta-functions with identical arguments on the critical line
- Zagier's 2-3-2 formula for multiple Eisenstein series
- Multiple Zeta Values in Positive Characteristic: A Kaneko-Zagier-Type Conjecture
- Kontsevich-Zagier conjecture for Hensel-minimal fields with sections
BibTeX
@misc{grozev2026weighted,
title = {{Weighted Generalizations of Zagier's Phenomenon}},
author = {Dragomir Grozev and Navid Safaei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24469},
eprint = {2609.24469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24469},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weighted Generalizations of Zagier's Phenomenon
AU - Dragomir Grozev
AU - Navid Safaei
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24469
UR - https://arxiv.org/abs/2609.24469
AB - We study sums associated with the action of PGL_2(ℤ) on continuous piecewise polynomial functions with exactly two real roots, both irrational. We form weighted sums of the positive parts of their normalized transforms, using nonnegative weights compatible with translation, reflection, and inversion. Under suitable continuity, finiteness, and convergence assumptions, we prove that these sums are well defined, bounded, $1$-periodic, and continuous on ℝ, and satisfy a reciprocal functional equation. We also extend the construction to finite families of distinct function orbits. Our framework recovers Zagier's constancy result and includes the full family of quadratic sums for which Bengoechea proved convergence.
ER -