Upper bounds for sums of powers of divisor functions (Q1185819)

The author gives sharp and uniform upper bounds for the sums \(D_ z(x,t)=\sum_{n\leq x}d_ z(n)^ t\), where \(d_ z(n)\) is the generalized divisor function defined by \(\zeta(s)^ z=\sum_{n\geq 1}d_ z(n)n^{-s}\). For example, Theorem 1.11 states that \[ D_ z(x,t)\leq x\exp\{(z^ t-1)\log\log x+z\log\log(3z)+O(z)\} \] holds uniformly for all real \(x\geq 3\), \(z>1\), and \(0<t\leq 1\). Similar, though more complicated estimates are given for the range \(t>1\). The result largely generalize and improve the classical estimates of \textit{K. K. Mardzhanishvili} [Dokl. Akad. Nauk SSSR 22, 387-389 (1939; Zbl 0021.20802)], especially with respect to the uniformity in the parameters \(z\) and \(t\). The author also gives a uniform upper bound for \(d_ z (n)\), from which he deduces an asymptotic formula for the maximal order of \(\log d_ z(n)\) when \(z=z(n)\) is allowed to vary with \(n\) in the range \(1+\varepsilon\leq z(n)\leq o(\log n)\).