Some Variants Of The Divisor Problems

Dirichlet divisor function d(n) =(?) and the summatory function D(x) =(?)are arithmetic functions in number theory, which occur in the study of many important problems,and there are many results about them. This thesis will study two variants of them as follows.Let d(n; r1,q1,r2,q2)be the number of factorization n ?nln2 satisfying ni?ri (mod qi)(i = 1,2) and A(x; r1,q1,r2,q2) be the error term of the summatory function of d(n; r1,q1,r2, q2)with x?q1q2, 1?ri?qi,and (ri,qi) = 1 (i = 1,2). We study the sign changes of?(x; r1,q1,r2,q2),and prove that for a sufficiently large constant C,?(q1q2x;r1,q1,r2,q2)changes sign in the interval [T, T + C(?)] for any large T. Meanwhile, we show that for a small posi-tive constant C, there exist infinitely many subintervals of length c'(?)log-7 T in [T, 2T] where±A(q1q2x;r1,q1,q2)>c5x1/4 always holds.Let S(x;a1/q1,a2/q2) =?'mn?cos(2?ma1/q1) sin (27?na2/q2), where x ? q1q2,1 ?ai ? qi,(ai,qi) =1(i=1,2). By connecting it with the divisor problem with congruence conditions, we establish an upper bound, mean-value, mean-square and some power-moments. Then, We study the sign changes of S(x;q1/q2,a2/q2), and prove that for a sufficiently large constant C, S(x;a1/q1,a2/q2)changes sign in the interval [T, T + C(?)] for any large T. Meanwhile, we show that for a s-mall constant c', there exist infinitely mady subintervals of length c'(?)log-7 T in [T, 2T] where±S(x;a1/q1,a2/q2)> C5(q1q2)3/4x1/4 always holds.