Gaussian Primes in Narrow Sectors

Representing ideal classes of ray class groups by product of prime ideals of small size

We prove that, for every modulus $\mathfrak{q}$, every class of the narrow ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field $\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$ of degree one with $\mathfrak{N}\mathfrak{p}\le (t(\mathbf{K})\mathfrak{N}\mathfrak{q})^3$, where $t(\mathbf{K})$ is an explicit function of $\mathbf{K}$ described in the paper. To achieve this result, we first obtain a sharp explicit Brun-Titchmarsh Theorem for ray classes and then an equally explicit improved Brun-Titchmarsh Theorem for large subgroups of narrow ra…

Zero-free angular sectors and lens-shaped regions for polynomials, with applications to irreducibility

For a real polynomial $f$ we present explicit zero-free angular sectors in the complex plane, symmetric with respect to the real axis, with angles depending only on the degree of $f$, and vertices expressed in terms of the coefficients of $f$. We also describe zero-free lens-shaped regions for $f$ that are associated to the zero-free sectors of the reciprocal of $f$. As an application, we use these zero-free regions to obtain irreducibility criteria for polynomials with integer coefficients that take a prime or a prime power value.

Sectorial equidistribution of the roots of $x^2 + 1$ modulo primes

The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec from 1995. We show that a subsequence of the roots of the equation remains equidistributed even when one adds a restriction on the primes which has to do with the angle in the plane formed by their corresponding representation as a sum of squares. Similar to Duke, Friedlander and Iwaniec, w…

Narrow progressions in the primes

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $δ>0$, one can find a "polynomial progression" $a+P_1(r),\dots,a+P_k(r)$ in $A$ with $0 < |r| \leq N^{o(1)}$, if $N$ is sufficiently large depending on $k,P_1,\dots,P_k$ and $δ$. In this paper we shorten the size of this progression to $0 < |r| \leq \log^L N$, where $L$ depends on $k,P_1,\dots,P_k$ and $δ$. In the linear case $P_i = (i-1)\mathbf{m}$, we can take $L$ independent…

$\mathbb Z[i]$ 的收缩扇形上的遍历平均

Ergodic Averages over Shrinking Sectors of $\mathbb Z[i]$

本文研究高斯整数收缩角扇形上的遍历平均，证明唯一遍历系统中连续函数平均值一致收敛，并给出有限生成乘法作用的定量比较，推论包括角位置独立性、高斯刘维尔相消及素数因子计数的联合等分布。

签名$(2p, 2q, r)$且窄类数为奇数的有效广义Fermat方程

Effective Generalized Fermat equation of signature $(2p, 2q, r)$ with odd narrow class number

本文研究签名$(2p,2q,r)$的广义Fermat方程，通过S-单位方程条件和模性证明，在窄类数为奇数时，对充分大的p,q无非平凡本原整数解，且常数V_r有效可计算。

射线类群中素理想的乘积

Products of prime ideals in ray class groups

本文证明在窄射线类群中，每个类可由三个范数不超过$(N\mathfrak q)^{103/64+\kappa}$的素理想乘积表示，改进了之前的$O_K((N\mathfrak q)^3)$界，并证明正比例的射线类可由两个素理想乘积表示。

Gaussian ideals在扇形和短区间中的Halász定理

Halász theorems for Gaussian ideals in sectors and short intervals

本文研究了在Gaussian理想上乘法函数的Halász定理，通过控制与Archimedean字符N^{it}的pretentious距离来获得界，同时证明了在扇形中的非pretentious性下，固定扇形内理想上f的和渐近等于无限制和的期望比例，并在非退化条件和共轭素数对条件下证明了扇形短区间中的Halász定理。

Gaussian Primes上的均值：Goldbach猜想与改进估计

Averages over the Gaussian Primes: Goldbach's Conjecture and Improving Estimates

本文证明了任意扇形区域内Gaussian primes的Goldbach猜想版本，包括三素数求和表示及二进制Goldbach猜想的密度版本。