In this paper, we construct new families of imaginary and real quadratic fields that are $p$-rational. In the imaginary case, we prove that for any positive integer $k$ and any integer $m$, the imaginary quadratic field $\mathbb{Q}\left(\sqrt{-(kp+m)}\right)$ is $p$-rational for sufficiently large primes $p$. The proof relies on Louboutin's bound on the class numbers of imaginary quadratic fields. As a corollary, we recover the $p$-rationality of consecutive quadratic fields, a result due to Chattopadhyay, Laxmi and Saikia \cite{CLS}. In the real case, we give an explicit proof of the $p$-rationality of the real quadratic field $\mathbb{Q}\left(\sqrt{p(p+1)}\right)$ for any odd prime $p$, and obtain new pairs of real quadratic fields $\left(\mathbb{Q}\left(\sqrt{p(p-2)}\right),\mathbb{Q}\left(\sqrt{p(p-1)}\right)\right)$ and $\left(\mathbb{Q}\left(\sqrt{p(p+1)}\right),\mathbb{Q}\left(\sqrt{p(p+2)}\right)\right)$ for any prime $p>3$. We also construct new examples of $p$-rational triquadratic fields.
Let $k=\mathbb{F}_q$, $E=\mathbb{F}_{q^n}$ and $\mathrm{Tr}=\mathrm{Tr}_{E/k}$. For $r\ge 2$, $a\in k^{\times}$ and $x\in E^{\times}$, let $\mathrm{N}(E,r,x,a)$ be the number of $r$-tuples $(x_1,\cdots,x_r)$ in $(E^{\times})^r$ satisfying $x_1\cdots x_r=x$ and $\mathrm{Tr}(x_1+\cdots+x_r)=a$. We prove $\left|\mathrm{N}(E,r,x,a)-\left((q^n-1)^{r-1}+(-1)^r\right)/q\right|\le (r^n-1) q^{\frac{(r-1)n-1}{2}}$. This proves the square-root estimate predicted in Wan's conjecture and generalizes a previous result of Moisio and Wan. For a finite semisimple algebra $B=\prod\limits_{i=1}^s M_{d_i}(\mathbb{F}_{q^{n_i}})$ over $k$ and a regular element $x\in B^{\times}$, the same method combined with Zelingher's formula leads to analogous square-root estimates.
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.
Xuejun Guo, Chen Lin, Zhefeng Xu· 0 citations
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