Let ?(x) denote the error term in the classical Dirichlet divisor problem,
and let the modified error term in the divisor problem be ?*(x) = -?(x) +
2?(2x)-1/2?(4x). We show that ?T+H,T ?*(t/2?)|?(1/2+it)|2dt<< HT1/6log7/2 T (T2/3+? ? H = H(T) ? T), ?T,0 ?(t)|?(1/2+it)|2dt <<
T9/8(log T)5/2, and obtain asymptotic formulae for ?T,0 (?*(t/2?))2|?(
1/2+it)|2 dt, ?T0 (?*(t/2?))3|?(1/+it)|2 dt. The importance of
the ?*-function comes from the fact that it is the analogue of E(T), the
error term in the mean square formula for |?(1/2+it)|2. We also show, if
E*(T) = E(T)-2??*(T/(2?)), ?T0 E*(t)Ej(t)|?(1/2+it)|2 dt << j,?
T7/6+j/4+? (j=1,2,3).