ON SOME MEAN VALUE RESULTS FOR THE ZETA-FUNCTION AND A DIVISOR PROBLEM

Abstract

Let $\Delta(x)$ denote the error term in the classical Dirichlet
divisor problem, and let the modified error term in the
divisor problem be $\Delta^*(x) =
 -\Delta(x)  + 2\Delta(2x) - \frac{1}{2}\Delta(4x)$. We show that
 $$
 \int_T^{T+H}\Delta^*\bigl(\frac{t}{2\pi}\bigr)|\zt|^2\,dt \;\ll\; HT^{1/6}\log^{7/2}T
 \quad(T^{2/3+\e} \le H = H(T) \le T),
 $$
 $$
 \int_0^T\Delta(t)|\zt|^2\,dt \;\ll\; T^{9/8}(\log T)^{5/2},
 $$
 and obtain  asymptotic formulae for
 $$
 \int_0^T{\Bigl(\Delta^*\bigl(\frac{t}{2\pi}\bigr)\Bigr)}^2|\zeta(1/2+it)|^2\,dt,\quad
 \int_0^T{\Bigl(\Delta^*\bigl(\frac{t}{2\pi}\bigr)\Bigr)}^3|\zeta(1/2+it)|^2\,dt.
 $$
 The importance of the $\Delta^*$-function comes from the fact that it is the analogue
 of $E(T)$, the error term in the mean square formula for $|\zeta(1/2+it)|^2$. We also show, if
 $E^*(T) = E(T) - 2\pi \Delta^*(T/(2\pi))$,
 $$
\int_0^T E^*(t)E^j(t)|\zeta(1/2+it)|^2\d t \;\ll_{j,\e}\; T^{7/6+j/4+\e}\quad(j= 1,2,3).
$$