On the Fractional Parts of Polynomials Modulo $p$

Xuejun Guo; Chen Lin; Zhefeng Xu

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arXiv · 2607.21259

On the Fractional Parts of Polynomials Modulo $p$

Xuejun Guo·Chen Lin·Zhefeng Xu

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Abstract

We study a half-interval distribution problem for polynomial residues modulo an odd prime $p$: how often the fractional part of $\varphi(x)/p$ lies in the upper half of the unit interval as $x$ ranges over $1\leq x< p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\#\left\{1\leq x< p/2:\left\{{\varphi(x)}/{p}\right\}>\frac12\right\} =\frac{p}{4}+O_\varphi(\sqrt p\log^2 p). $ We then show that the error term can be improved to $O_\varphi(\sqrt p\log p)$ for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials $\varphi(x)=x^m$, we further obtain the bound $O_m(\sqrt p\log\log p)$ under the Generalized Riemann Hypothesis. Finally, in the case $m=2$, we prove an unconditional matching lower bound, showing that the factor $\log\log p$ is best possible in this setting.

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Xuejun Guo, Chen Lin, Zhefeng Xu. 2026-07-23. On the Fractional Parts of Polynomials Modulo $p$. https://arxiv.org/abs/2607.21259

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