For the ordinary \(m\)-dimensional divisor function \(d_ m(n)\), \textit{J. M. De Koninck} and \textit{A. Ivić} [Topics in arithmetical functions. Amsterdam etc.: North-Holland (1980; Zbl 0442.10032)] proved that \[ \sum_{n\leq x}\frac{1}{d_ m(n)}= x\sum^{N}_{k=0}A_ k^*(\log x)^{1/m-k-1}+O_ N(x(\log x)^{1/m-N-2}) \tag{1} \] where \(x\) is a large real variable, \(N\) is an arbitrary (fixed) positive integer and the \(A_ k^*\)'s are computable constants. In the present paper, this problem is considered in the general context of divisor functions in algebraic number fields which includes the sums \(\sum_{n\leq x}1/d_ m(n)\) and \(\mathop{{\sum}'}_{n\leq x} 1/r(n)\) as special cases. (Here \(r(n)\) is the usual notation for the number of ways to write \(n\in\mathbb N\) as a sum of two integer squares and \(\sum '\) means that terms which formally would read \(1/0\) are to be omitted.) The estimate is sharpened to an extent which seems to be ``final'', at the present state of the art, in view of the known zero-free region of the (Dedekind) zeta-function. To be precise, for \(m\geq 1\) and a Galois extension \(K\) of the rationals of degree \([K : \mathbb Q]=r\geq 1\), the arithmetic function \[ d_{m,K}(n)=\#\{({\mathcal A}_ 1,...,{\mathcal A}_ m)\in {\mathcal R}^ m:\;N({\mathcal A}_ 1...{\mathcal A}_ m)=n\} \] is considered, where \(N(\cdot)\) denotes the norm in the ring \({\mathcal R}\) of integer ideals \(\neq (0)\) of \(K\), as usual. The main result of the article is the asymptotic formula \[ \mathop{{\sum}'}_{n\leq x} \frac{1}{d_{m,K}(n)}= x\sum^{M(x)}_{k=0}A_ k(\log x)^{-k-1+1/mr^ 2}+O(x\exp (-c(\log x)^{3/5}(\log \log x)^{-1/5})) \] with \(M(x)=[a(\log x)^{3/5}(\log \log x)^{-6/5}]\), computable coefficients \(A_ k\) and positive constants \(a\) and \(c\), all of them depending on \(K\) and \(m\). As a corollary, this contains not only an improvement of (1), but also the expansion \[ \mathop{{\sum}'}_{n\leq x} \frac{1}{r(n)}=\frac{x}{(\log x)^{3/4}}\sum^{M(x)}_{k=0}A_ k^{**}(\log x)^{-k}+O(x \exp (- c(\log x)^{3/5}(\log \log x)^{-1/5})), \] where, numerically, \(A_ 0^{**}=0,122....\) Finally, an analogous improvement is obtained for the (generalized) Selberg divisor problem which concerns the average order of the coefficients in the Dirichlet series for \((\zeta_ K(s))^{\alpha}\) where \(\zeta_ K(s)\) is the Dedekind zeta-function of an arbitrary algebraic number field K, and \(\alpha\in\mathbb R\setminus\mathbb Z\). (Cf. \textit{A. Selberg}, J. Indian Math. Soc., N. Ser. 18, 83--87 (1954; Zbl 0057.28502)]).