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

Abstract

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $ω\subset \mathbb{T}$. There is an integer $N_ω$, so that every odd integer $n$ with $N(n)>N_ω$ and $\text{dist}( \text{arg}(n) , \mathbb{T}\setminus ω) > (\log N(n)) ^{-B}$, is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with $\text{arg}(p_j) \in ω$, for $j=1,2,3$. A density version of the binary Goldbach conjecture in a sector is also proved.

Figures (6)

• Figure 1: The lattice points in the box $B_q$, for $q = 2+i$. There are exactly $N(q) = 5$ points inside the box.
• Figure 2: The box $B_q$ for $q = 4 + 2i$. Note that $4+2i = 2(2+i)$, hence $B_{4+2i}$ is partitioned by $2^2$ Gaussian translations of $B_{2+i}$.
• Figure 3: A point $x$ is in the sector $S _{\omega }$. The parallelogram $T^\omega (x)$ is pictured inside the sector.
• Figure 4: The distances $\lVert \frac{ra}{q}+hq\beta+r\beta\rVert$ for a fixed $h$ are essentially uniformly distributed in case 3.
• Figure 5: The decomposition of the complex field.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.5
• Lemma 2.7
• proof
• Proposition 2.16
• proof
• Lemma 2.20
• proof
• Lemma 3.1