On some mean value results for the zeta-function and a divisor problem

Abstract: Let

D e l t a (x)

denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be

D e l t a^{*} (x) = - D e l t a (x) + 2 D e l t a (2 x) - f r a c 12 D e l t a (4 x)

. We show that

int_T^{T+H}Delta^*�igl(frac{t}{2pi}�igr)|zeta(1/2+it)|^2dt ;ll; HT^{1/6}log^{7/2}T quad(T^{2/3+varepsilon} le H = H(T) le T),

int_0^TDelta(t)|zeta(1/2+it)|^2dt ;ll; T^{9/8}(log T)^{5/2},

and obtain asymptotic formulae for

int_0^T{Bigl(Delta^*�igl(frac{t}{2pi}�igr)Bigr)}^2 |zeta(1/2+it)|^2dt,quad int_0^T{Bigl(Delta^*�igl(frac{t}{2pi}�igr)Bigr)}^3|zeta(1/2+it)|^2dt.

The importance of the

D e l t a^{*}

-function comes from the fact that it is the analogue of

E (T)

, the error term in the mean square formula for

| z e t a (1 / 2 + i t) |^{2}

. We also show, if

E^{*} (T) : = E (T) - 2 p i D e l t a^{*} (T / (2 p i))

,

int_0^T E^*(t)E^j(t)|zeta(1/2+it)|^2dt ; ll_{j,varepsilon}; T^{7/6+j/4+varepsilon}quad(j= 1,2,3).