LARGE BIAS FOR INTEGERS WITH PRIME FACTORS IN ARITHMETIC PROGRESSIONS. Issue 1 (15th February 2018)

• Record Type: Journal Article
• Authors: Meng, Xianchang
• Is Part Of: Mathematika. Volume 64:Issue 1(2018)
• Page Start: 237
• Page End: 252
• Publication Date: 2018-02-15
• Subjects: 11M06 -- 11N13 -- 11N69 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579317000584
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract : We prove asymptotic formulas for the number of integers at most x that can be written as the product of k ( ⩾ 2 ) distinct primes p 1 ⋯ p k with each prime factor in an arithmetic progression p j ≡ a j mod q, ( a j, q ) = 1 ( q ⩾ 3, 1 ⩽ j ⩽ k ) . For any A > 0, our result is uniform for 2 ⩽ k ⩽ A log log x . Moreover, we show that there are large biases toward certain arithmetic progressions ( a 1 mod q, …, a k mod q ), and such biases have connections with Mertens' theorem and the least prime in arithmetic progressions.