Let \(\sigma_a(n) = \sum_{d| n}d^a\) and let, for \(-1 < a < 0,\, x\geq1\) \[ \Delta_a(x) := {\sum_{n\leq x}}' \sigma_a(n) - \zeta(1-a)x - {\zeta(1+a)\over1+a}x^{1+a} + {\textstyle{1\over2}}\zeta(-a), \] where the dash \('\) means that the last term in the sum is to be halved if \(x\) is an integer. The limiting case of \(\Delta_a(x)\), as \(x\to0-\), is \(\Delta(x)\), the error term in the classical Dirichlet divisor problem. The author uses a weighted Voronoi type formula for \(\Delta_a(x)\) of T. Meurman [Acta Arith. 74, 351–364 (1996; Zbl 0848.11043)] to establish the mean square bound \[ \int_T^{2T}(\Delta_a(t+h)-\Delta_a(t))^2\, dt \ll_\delta Th^{1+2a}\min\left({1\over{1\over2}-| a| },\,\log h\right), \tag{1} \] where the \(\ll\)-constant depends only on a small, positive number \(\delta\), \(a\in [-{1\over2}.\,\delta]\), \(\,1\ll h \leq \sqrt{T}\). This result is new when \(a = -1\), and in the remaining range it was obtained by I. Kiuchi and Y. Tanigawa [Arch. Math. 71, 445–534 (1998; Zbl 0926.11072)]. The author uses (1) to deduce a result about intervals where \(\Delta_a(x)\) has no sign changes. He also deduces, from a general result of A. Ivić [Acta Arith. 56, 135–159 (1990; Zbl 0659.10053)], that \(\Delta_a(x)\) has a sign change in \([T, T + c_a\sqrt{T}\,]\) for some constant \(c_a > 0\) if \(-{1\over2} \leq a \leq 0\) and sufficiently large \(T\).