On the Fractional Parts of Polynomials Modulo p
Abstract
We study a half-interval distribution problem for polynomial residues modulo an odd prime p: how often the fractional part of φ(x)/p lies in the upper half of the unit interval as x ranges over 1≤ x< p/2. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\#\left\{1\leq x< p/2:\left\{{φ(x)}/{p}\right\}>\frac12\right\} =\frac{p}{4}+O_φ(\sqrt p\log^2 p). $ We then show that the error term can be improved to Oφ(√plog p) for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials φ(x)=xm, we further obtain the bound Om(√ploglog p) under the Generalized Riemann Hypothesis. Finally, in the case m=2, we prove an unconditional matching lower bound, showing that the factor loglog p is best possible in this setting.
Versions
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BibTeX
@misc{guo2026fractional,
title = {{On the Fractional Parts of Polynomials Modulo $p$}},
author = {Xuejun Guo and Chen Lin and Zhefeng Xu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.21259},
eprint = {2607.21259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21259},
}
RIS
TY - UNPB
TI - On the Fractional Parts of Polynomials Modulo p
AU - Xuejun Guo
AU - Chen Lin
AU - Zhefeng Xu
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2607.21259
UR - https://arxiv.org/abs/2607.21259
AB - We study a half-interval distribution problem for polynomial residues modulo an odd prime p: how often the fractional part of φ(x)/p lies in the upper half of the unit interval as x ranges over 1≤ x< p/2. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\#\left\{1\leq x< p/2:\left\{{φ(x)}/{p}\right\}>\frac12\right\} =\frac{p}{4}+O_φ(\sqrt p\log^2 p). $ We then show that the error term can be improved to O_φ(√plog p) for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials φ(x)=x^m, we further obtain the bound O_m(√ploglog p) under the Generalized Riemann Hypothesis. Finally, in the case m=2, we prove an unconditional matching lower bound, showing that the factor loglog p is best possible in this setting.
ER -