Bibliographic Details
| Title: |
Algebraic analogues of results of Alladi–Johnson using the Chebotarev Density Theorem: Algebraic analogues of results of Alladi–Johnson...: S. Sengupta. |
| Authors: |
Sengupta, Sroyon1 (AUTHOR) sengupta.s@ufl.edu |
| Source: |
Research in Number Theory. 3/19/2025, Vol. 11 Issue 2, p1-18. 18p. |
| Subject Terms: |
*PRIME factors (Mathematics), *PRIME number theorem, *PRIME numbers, *MOBIUS function, *FINITE groups, *CONJUGACY classes |
| Abstract: |
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 ; p 1 (n) ≡ ℓ (m o d k) μ (n) ω (n) n = 0 , where μ (n) is the Möbius function, ω (n) is the number of distinct prime factors of n, and p 1 (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 p 1 (n) = C μ (n) ω (n) n = 0. where K / Q p 1 (n) is the Artin symbol. When K is a cyclotomic extension of Q , this reduces to the exact case of A–J's result. [ABSTRACT FROM AUTHOR] |
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