2019/03/14 by Theophilus Agama, Agama, Theophilus
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #General Mathematics (math.GM) #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1903.09257
openalex publication_date 2019/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we develop a general method for estimating correlations of the forms ∑ n≤ xG(n)G(x-n)and ∑ n≤ xG(n)G(n+l)for a fixed 1≤ l≤ x and where G:ℕ\longrightarrow ℝ+. To distinguish between the two types of correlation, we call the first correlation the type 2 correlation and the second the type 1 correlation. As an application, we estimate the lower bound for the type 2 correlation of the master function ∑ n≤ xΥ(n)Υ(n+l0)≥ (1+o(1))(x)/(2C(l0))log log 2xprovided that Υ(n)Υ(n+l0)>0. We also use this method to provide a first proof of the twin prime conjecture showing that ∑n≤ xΛ(n)Λ(n+2)≥ (1+o(1))(x)/(2C(2))for some C:=C(2)>0.