On Chen's theorem, Goldbach's conjecture and almost prime twins II (2405.05727v4)

Abstract: Let $N$ denote a sufficiently large even integer and $x$ denote a sufficiently large integer, we define $D_{1,2}(N)$ as the number of primes $p$ that such that $N - p$ has at most 2 prime factors. In this paper, we show that $D_{1,2}(N) \geqslant 1.9728 \frac{C(N) N}{(\log N)^2}$, which is rather near to the asymptotic constant $2$ in Hardy--Littlewood conjecture for Goldbach's conjecture. We also get similar results on twin prime problem and additive representations of integers. The proof combines various techniques in sieve methods, such as weighted sieve, Chen's switching principle, new distribution levels proved by Lichtman and Pascadi, Chen's double sieve and Harman's sieve.

• The paper refines Chen's theorem by establishing an improved lower bound of 1.9728 for primes in Goldbach's conjecture, significantly surpassing previous results.
• It extends the refinement to twin prime problems by proving a new lower bound for primes p such that p+2 has at most two prime factors.
• Advanced weighted sieve methods, including Chen’s switching principle and Buchstab's identity, are employed alongside novel distribution levels to optimize the estimates.

Refinement of Chen's Theorem and Related Problems

This paper (2405.05727) advances the study of Goldbach's conjecture and twin prime problems, building upon Chen's theorem. It introduces improvements to the lower bounds in Chen's theorem, the almost prime twins problem, and additive representations of integers by using refined sieve methods and distribution levels.

Improved Lower Bound for Goldbach's Conjecture

The paper focuses on the quantity $D_{1,2}(N)$, which represents the number of primes $p$ such that $N-p$ has at most two prime factors. The main result is the following theorem:

Theorem 1.1:

$$D_{1,2}(N) \geqslant 1.9728 \frac{C(N) N}{(\log N)^2},$$

where $C(N)$ is the singular series defined as:

$$C(N) = \prod_{\substack{p \mid N \ p>2}} \frac{p-1}{p-2} \prod_{p>2}\left(1-\frac{1}{(p-1)^{2}}\right).$$

This result improves upon the previous bound of $1.733$ obtained by the author in a prior work (2405.05727) and significantly refines Wu's earlier record of $0.899$. The new constant is close to the conjectured asymptotic constant $2$ for $D_{1,1}(N)$, which counts the number of primes $p$ such that $N-p$ is also prime. The paper also extends this result to additive representations of integers. For relatively prime square-free positive integers $a, b$, the quantity $R_{a,b}(M)$ represents the number of primes $p$ such that $ap$ and $M-ap$ are both square-free, $b \mid (M-ap)$, and $\frac{M-ap}{b}=P_2$. The following theorem is established:

Theorem 1.2:

$$R_{a, b}(M) \geqslant 1.9728 \frac{C(abM) M}{a b(\log M)^{2}}.$$

Advancement in the Twin Prime Problem

The paper also considers the twin prime problem, specifically the quantity $\pi_{1,2}(x)$, which counts the number of primes $p \leqslant x$ such that $p+2$ has at most two prime factors. The following theorem is presented:

Theorem 1.3:

$$\pi_{1,2}(x) \geqslant 1.2759 \frac{C_2 x}{(\log x)^2},$$

where $C_2$ is defined as:

$$C_2 = 2 \prod_{p>2}\left(1-\frac{1}{(p-1)^{2}}\right).$$

This improves on the previous bound of $1.238$ from the author's prior work (2405.05727).

Distribution Levels and Sieve Methods

The proofs rely on sieve methods, including weighted sieves, Chen's switching principle, Chen's double sieve, and Harman's sieve. The improvements are achieved through the incorporation of new distribution levels for primes, derived from the works of Lichtman (Lichtman, 2023) and Pascadi (Pascadi, 1 May 2025). These distribution levels provide better control over the error terms in the sieve estimates. The paper uses the Kim-Sarnak bound, $\theta_1 = \frac{7}{32}$, and defines functions $\boldsymbol{\vartheta}_{\alpha}(t_{1})$ and $\boldsymbol{\vartheta}_{\alpha}(t_1, t_2, t_3)$ with $\alpha=0 \text{ or } 1$, similar to those in Lichtman's work (Lichtman, 2023).

Lemmas 2.1 and 2.2 present distribution results used for Theorem 1.1, based on Lichtman's work (Lichtman, 2023), with a distribution level of $\frac{19101}{32000} \approx 0.5969$. Lemmas 2.3 and 2.4 provide distribution results for Theorem 1.3, based on Pascadi's work (Pascadi, 1 May 2025), with a distribution level of $\frac{2497}{4000} = 0.62425$. These lemmas provide estimates for the distribution of primes in arithmetic progressions, which are essential for the sieve methods used in the proofs.

Weighted Sieve Setup

The paper employs a weighted sieve method to obtain the lower bounds for $D_{1,2}(N)$ and $\pi_{1,2}(x)$. Lemmas 3.1 and 3.2 give the weighted sieve setups for Goldbach's conjecture and the twin prime problem, respectively. These lemmas decompose the quantities $D_{1,2}(N)$ and $\pi_{1,2}(x)$ into several terms involving the sieve function $S(\mathcal{A}; \mathcal{P},z)$ and its variants. The weights are chosen to optimize the lower bounds.

Buchstab's Identity and Chen's Switching Principle

The proofs involve extensive manipulations using Buchstab's identity and Chen's switching principle. Buchstab's identity is used to decompose the sieve function into sums over primes in certain ranges, allowing for a more refined analysis. Chen's switching principle is applied to handle certain terms in the weighted sieve, which involves switching the roles of the variables to obtain better estimates.

Numerical Computations

The paper relies on numerical computations to evaluate various integrals and functions that arise in the sieve estimates. These computations are crucial for obtaining the explicit constants in the lower bounds. The values of the functions $F(s)$, $f(s)$, $\omega(u)$, $H(s)$, and $h(s)$ are used in the numerical calculations.

Conclusion

This paper (2405.05727) makes a significant contribution to the study of Goldbach's conjecture and twin prime problems. The improved lower bounds for $D_{1,2}(N)$ and $\pi_{1,2}(x)$ represent a substantial advancement in the field. The methods used in the paper, including the incorporation of new distribution levels and the application of Chen's switching principle, are of independent interest and may have further applications in number theory.