Mean square of the remainder term in the Dirichlet divisor problem (Q1909865)

If \(\Delta (x)\) is the error term in the Dirichlet divisor problem, then it was shown by \textit{K.-C. Tong} [Acta Math. Sin. 6, 515-541 (1956; Zbl 0075.25003)]\ that \(\int^X_2 \Delta (x)^2 dx= cX^{3/2}+ F(X)\) for an appropriate constant \(c\), where \(F(X)\ll X\log^5 X\). It has been conjectured that \(F(X)\ll X^{3/4+ \varepsilon}\) for any \(\varepsilon>0\), which would imply that \(\Delta (x)\ll x^{1/4+ \varepsilon}\). However it is shown in the present paper that \[ \int^X_2 F(x) dx=- (8\pi^2 )^{-1} X^2\log^2 X+ c_1 X^2\log X+ O(X^2) \] for a certain constant \(c_1\). Thus, in particular, \(F(X)= \Omega_- (X\log^2 x)\). The proof gives a careful analysis of \(F(X)\) which ultimately depends on the asymptotic formula for \(\sum_{m\leq x} d(m) d(m+ h)\).