The function \(\tau (n,\theta)=\sum_{d\mid n}d^{i\theta}\) \((\theta \in\mathbb{R})\) occurs in a natural way in the study of various problems in number theory, such as those concerning the distribution of the divisors of \(n\). In chapter 3 of [the author and G. Tenenbaum, Divisors. Cambridge etc.: Cambridge University Press (1988; Zbl 0653.10001)], the question is raised whether there exists \(\lambda = \lambda (a.b)\), where \(0<a<b\), such that
\[ \max_{a\leq \theta \leq b}| \tau (n,\theta)| <(\log n)^{\lambda +\varepsilon}\;p.p. \tag{*} \]
This problem is considered in the present paper in the case when \(b=a+(\log n)^{K-1}\) and \(a\) is fixed; it is shown that for each \(K\), there exists \(\lambda =\lambda (K)<\log 2\) such that (*) holds. For \(K\leq 0\), \(\lambda (K)=0\), and for \(K>0\), \(\lambda(K)\) is given explicitly in terms of the unique solution of a certain differential equation involving \(K\). For positive \(K\), the inequalities \[ \lambda (K)\leq \log 2-(\pi /4e)e^{-2K}\text{ and } \lambda (K)\leq \pi (K/6)^{1/2} \] are obtained, the second being stronger than the first when \(K\) is small.