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 $\varphi(x)/p$ 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.