2024/10/29 by Sengupta, Sroyon · 1 citation
Mathematics · #FOS: Mathematics #Functional Equations Stability Results #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2410.22226
openalex publication_date 2024/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
\small We aim to get an algebraic generalization of Alladi-Johnson's (A-J) work on Duality between Prime Factors and the Prime Number Theorem for Arithmetic Progressions - II, using the Chebotarev Density Theorem (CDT). It has been proved by A-J, that for all positive integers k,ℓ such that 1≤ ℓ≤ k and (ℓ,k)=1, ∑n≥ 2; p1(n) ≡ ℓ (mod k)(μ(n)ω(n))/(n) = 0, \textit\small where μ(n) is the Möbius function, ω(n) is the number of distinct prime factors of n, and p1(n) is the smallest prime factor of n. In our work here, we will prove the following result: If C is a conjugacy class of the Galois group of some finite extension K of ℚ, then ∑ n ≥ 2; [(K/ℚ)/(p1(n))]=C (μ(n)ω(n))/(n) = 0. \textit\small where [\fracK/ℚp1(n)] is the Artin symbol. When K is a cyclotomic extension of ℚ, this reduces to the exact case of A-J's result.