Summatory function of the number of prime factors

2018/01/21 by Xianchang Meng, Meng, Xianchang
Mathematics · Computer Science · #Analytic Number Theory Research #Coding theory and cryptography #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1801.06906

Abstract

We consider the summatory function of the number of prime factors for integers ≤ x over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg Martin conjectured that the difference of the summatory functions should attain a constant sign for all sufficiently large x. In this paper, we provide strong evidence for Greg Martin's conjecture. Moreover, we derive a general theorem for arithmetic functions from the Selberg class.

Related

• Duality between prime factors and the Prime Number Theorem for Arithmetic Progressions -- II
• On sums of prime factors
• On the proximity of multiplicative functions to the function counting prime factors with multiplicity
• Dirichlet's Theorem and Jacobsthal's Function
• New bounds and progress towards a conjecture on the summatory function of (-2)^Ω(n)