On the Divisor Problem with Congruence Conditions  

带同余条件的除数问题

作　　者：JIA Lirui ZHAI Wenguang CAI Tianxin 贾丽慈;翟文广;蔡天新(北京交通大学数学与统计学院,北京,100044;清华大学数学科学系,北京,100084;中国矿业大学(北京)理学院,北京,100083;浙江大学数学科学学院,杭州,浙江,310027)

机构地区：[1]Department of Mathematics,Beijing Jiaotong University,Beijing,100044,P.R.China [2]Department of Mathematical Sciences,Tsinghua University,Beijing,100084,P.R.China [3]Department of Mathematics,China University of Mining and Technology(Beijing),Beijing,100083,P.R.China [4]School of Mathematical Sciences,Zhejiang University,Hangzhou,Zhejiang,310027,P.R.China

出　　处：《数学进展》2025年第1期1-17,共17页Advances in Mathematics(China)

基　　金：supported by the Talent Fund of Beijing Jiaotong University(No.2020RC012);NSFC(No.11871295),supported by NSFC(No.11971476),supported by NSFC(No.12071421)。

摘　　要：Let d(n;r_(1),q_(1),r_(2),q_(2))be the number of factorization n=n_(1)n_(2)satisfying n_i≡r_i(mod q_i)(i=1,2)andΔ(x;r_(1),q_(1),r_(2),q_(2))be the error term of the summatory function of d(n;r_(1),q_(1),r_(2),q_(2)).Suppose x≥(q_(1)q_(2))^(1+ε),1≤r_i≤q_i,and(r_i,q_i)=1(i=1,2).This paper studies the power moments and sign changes ofΔ(x;r_(1),q_(1),r_(2),q_(2)).We prove that for sufficiently large constant C,Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))changes sign in the interval[T,T+C√T]for any large T.Meanwhile,we show that for small constants c and c,there exist infinitely many subintervals of length c√log^(-7)T in[T,2T]where±Δ(q_(1)q_(2)x:r_(1),q_(1),r_(2),q_(2))>cx^(1/4)always holds.记d(n;r_(1),q_(1),r_(2),q_(2))为n=n_(1)n_(2)(n_(i)≡r_(i)(mod q_(i)),i=1,2)的表法个数,Δ(x;r_(1),<q_(1),r_(2),q_(2))为其和函数的余项.令x≥(q_(1)q_(2))^(1+ε),1≤r_(i)≤q_(i),且(r_(i),q_(i))=1(i=1,2),本文研究了Δ(x;r_(1),q_(1),r_(2),q_(2))的积分均值和符号变换.证明当常数C充分大时,对于任意充分大的常数T,Δ(q_(1)q_(2)x;r_(1),q_(1),r_(2),q_(2))在区间[T,T+C√T]内变号.同时还证明对于很小的常数c和c,区间[T,2T]内有无穷多个长度为c√Tlog^(-7)T的子区间,且在任一子区间内±△(q_(1)q_(2)x;r_(1),q_(1),r_(2),q_(2))>cx_(1/4)恒成立.

关 键 词：divisor problem sign change congruence condition 

分 类 号：O156.4[理学—数学]