发表机构
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出Zagier现象的重加权推广，研究PGL2(Z)作用下分段多项式函数的加权和，证明其有界、周期且满足倒数方程，并涵盖Bengoechea的收敛结果。
AI 中文摘要
我们研究与$PGL_2(\mathbb Z)$在具有恰好两个实根（均为无理数）的连续分段多项式函数上的作用相关的和。我们使用与平移、反射和反演相容的非负权重，对其归一化变换的正部形成加权和。在适当的连续性、有限性和收敛性假设下，我们证明这些和是良定义的、有界的、$1$-周期的且在$\mathbb R$上连续，并满足倒数函数方程。我们还将构造推广到有限个不同函数轨道的情形。我们的框架恢复了Zagier的常值结果，并包含了Bengoechea证明收敛的完整二次和族。
英文摘要
We study sums associated with the action of $PGL_2(\mathbb Z)$ 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 $\mathbb R$, 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.
Comments11 pages
