带同余条件的除数问题
数论
2019-08-16 v1
摘要
设 d ( n ; r 1 , q 1 , r 2 , q 2 ) d(n; r_1, q_1, r_2, q_2) d ( n ; r 1 , q 1 , r 2 , q 2 ) n i ≡ r i ( m o d q i ) n_i\equiv r_i\pmod{q_i} n i ≡ r i ( mod q i ) i = 1 , 2 i=1,2 i = 1 , 2 n = n 1 n 2 n=n_1n_2 n = n 1 n 2 Δ ( x ; r 1 , q 1 , r 2 , q 2 ) \Delta(x; r_1, q_1, r_2, q_2) Δ ( x ; r 1 , q 1 , r 2 , q 2 ) d ( n ; r 1 , q 1 , r 2 , q 2 ) d(n; r_1, q_1, r_2, q_2) d ( n ; r 1 , q 1 , r 2 , q 2 ) x ≥ ( q 1 q 2 ) 1 + ε , 1 ≤ r i ≤ q i x\geq (q_1q_2)^{1+\varepsilon}, 1\leq r_i\leq q_i x ≥ ( q 1 q 2 ) 1 + ε , 1 ≤ r i ≤ q i ( r i , q i ) = 1 (r_i, q_i)=1 ( r i , q i ) = 1 i = 1 , 2 i=1, 2 i = 1 , 2 Δ ( x ; r 1 , q 1 , r 2 , q 2 ) \Delta(x; r_1, q_1, r_2, q_2) Δ ( x ; r 1 , q 1 , r 2 , q 2 ) C C C T T T Δ ( q 1 q 2 x ; r 1 , q 1 , r 2 , q 2 ) \Delta(q_1q_2x; r_1, q_1, r_2, q_2) Δ ( q 1 q 2 x ; r 1 , q 1 , r 2 , q 2 ) [ T , T + C T ] [T,T+C\sqrt{T}] [ T , T + C T ] c ′ c' c ′ [ T , 2 T ] [T,2T] [ T , 2 T ] c ′ T log − 7 T c'\sqrt{T}\log^{-7}T c ′ T log − 7 T ± Δ ( q 1 q 2 x ; r 1 , q 1 , r 2 , q 2 ) > c 5 x 1 4 \pm \Delta(q_1q_2x; r_1, q_1, r_2, q_2)> c_5x^\frac{1}{4} ± Δ ( q 1 q 2 x ; r 1 , q 1 , r 2 , q 2 ) > c 5 x 4 1
引用
@article{arxiv.1908.05598,
title = {On the divisor problem with congruence conditions},
author = {Lirui Jia and Wenguang Zhai and Tianxin Cai},
journal= {arXiv preprint arXiv:1908.05598},
year = {2019}
}
备注
arXiv admin note: substantial text overlap with arXiv:1603.04977