arXiv subjects
Publications and source records attributed to Mizuki Akeno.
We prove that, for almost all even integers $N>0$, the set of Goldbach primes $\mathbb{P} \cap (N-\mathbb{P})$ has a level of distribution $1/6$. As applications, we show that almost all even integers $N>0$ can be written as the sum of two primes $p_1, p_2$ such that $p_1-p_2+1 \in \mathbb{P}_4$. We also prove an analogous result with $2p_1 p_2+1 \in \mathbb{P}_{13}$ for almost all integers $N>0$ with $6\mid N$.
In this paper, we obtain upper and lower bounds for the partition function $p(n)$ by using an elementary geometric inequality in Euclidean space, and we extend the method to generalizations of the partition function.
We study small gaps between Goldbach primes $\mathbb{P} \cap (N-\mathbb{P})$ using the Bombieri-Davenport method and the Maynard-Tao method, and compare the two. We show that for almost all even integers $N$, the smallest gap in $\mathbb{P} \cap (N-\mathbb{P})$ is at most $0.765\ldots$ times the average gap, using the Bombieri-Davenport method. This improves a recent result of Tsuda. We also demonstrate that a straightforward application of the Maynard-Tao method is insufficient to improve this bound. However, it allows us to establish the existence of bounded gaps between Goldbach primes with bounded error for almost all even integers $N$.