For \(y\geq 1\) let \[ \Delta_a(y):= \mathop{{\sum}'}_{n\leq y} \sigma_a(n)- \zeta(1- a) y- {\zeta(1+ a)\over 1+ a} y^{1+ a}+ {1\over 2} \zeta(- a),\;\sigma_a(n)= \sum_{d\mid n} d^a, \] where \(\mathop{{\sum}'}_{n\leq y}\) means that the last summand is halved if \(y\) is an integer. The author proves \[ \int^x_1 \Delta_a(y)^2 dy= \begin{cases} c_1 x^{3/2+ a}+ O(x) \quad & - 1/2< a< 0,\\ c_2 x\log x+ O(x) \quad & a= -1/2,\\ O(x) \quad & - 1< a< - 1/2,\end{cases} \] with explicitly given \(c_1, c_2> 0\). These results are the sharpest ones hitherto, and it may be reasonably conjectured that the \(O\)-terms above are in fact \(\Omega(x)\). The proofs use several intricate arguments, the key one being embodied in Lemma 1. It gives \(\Delta_a(y)\) \((- 1< a< 0)\) as a sum of two smoothed sums, plus error terms. It is the second of these sums, namely \[ R_a(y, X, Z):= {1\over 2\pi} \sum_{n\leq Z} \sigma_a(n) \int^2_1 \int^\infty_{u X} t^{- 1} \sin(4\pi(\sqrt y- \sqrt u) \sqrt t) dt du,\;X\geq y,\;Z\geq 2y, \] which accounts for the jumps of \(\Delta_a(y)\) at integers. Thus the easy estimate \(R_a(y, X, Z)\ll y^\varepsilon\) cannot be improved in general, but it can be improved if \(y\) is not near an integer and \(X\) is large.