发表机构
School of Mathematics, Nanjing University; Research Center for Number Theory and Its Applications, Northwest University（南京大学数学学院; 西北大学数论及其应用研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究模奇素数\(p\)时多项式余数的半区间分布问题，利用有限傅里叶展开和韦伊界证明渐近公式，改进误差项，对偶数单项式在广义黎曼假设下得更好界，\(m = 2\)时证明匹配下界及\(\log\log p\)最优。
AI 中文摘要
我们研究了模奇素数\(p\)的多项式余数的半区间分布问题：当\(x\)在\(1\leq x\lt p/2\)范围内时，\(\varphi(x)/p\)的分数部分位于单位区间上半部分的频率。利用有限傅里叶展开和韦伊界，我们证明了渐近公式\(\#\left\{1\leq x\lt p/2:\left\{\varphi(x)/p\right\}>\frac{1}{2}\right\}=\frac{p}{4}+O_{\varphi}(\sqrt{p}\log^{2}p)\)。然后表明，对于任意二次多项式和满足适当反射对称性的多项式，误差项可改进为\(O_{\varphi}(\sqrt{p}\log p)\)。对于偶数单项式\(\varphi(x)=x^{m}\)，在广义黎曼假设下进一步得到界\(O_{m}(\sqrt{p}\log\log p)\)。最后，在\(m = 2\)的情况下，我们证明了一个无条件的匹配下界，表明在这种情况下\(\log\log p\)因子是最优的。
英文摘要
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 $1\leq x< p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula $\#\left\{1\leq x< p/2:\left\{{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 $O_φ(\sqrt p\log p)$ for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials $φ(x)=x^m$, we further obtain the bound $O_m(\sqrt p\log\log p)$ 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.
Comments12 pages
