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The maximal size of the \(k\)-fold divisor function for very large \(k\). (English) Zbl 1465.11193

Let \(\tau_k\) be the \(k\)th Dirichlet-Piltz divisor function and set \(\tau:=\tau_2\) as usual. It has long been known [S. Wigert, Arkiv f. Mat., Astr. och Fys. 3, No. 18, 9 p. (1907; JFM 38.0249.01)] that \[\log \tau(n) \leqslant \left( 1 + o(1)\right) \frac{\log 2 \log n}{\log \log n}\] as \(n \to \infty\), and K. K. Norton [J. Number Theory 40, 60–85 (1992; Zbl 0748.11046)] proved that a similar inequality holds for \(\tau_k\) as long as \(k\) is allowed to vary with \(n\) in the range \(1 + \varepsilon \leqslant k \leqslant o (\log n)\). In the paper under review, the author studies the case \(\frac{k}{\log n} \to \kappa\) for a fixed \(\kappa > 0\), which enables him to correct a previous result announced by V. N. Chubarikov and G. V. Fedorov [Stud. Syst. Decis. Control 30, 29–36 (2015; Zbl 1335.11085)]. If \(s > 1\) is implicitely defined by \[\sum_p \frac{\log p}{p^s-1} = \frac{1}{\kappa}\] then the author shows that \[\log \tau_k(n) \leqslant \left( s + \kappa \sum_p \sum_{\alpha = 1}^\infty \frac{1}{\alpha ^{\alpha s}} + o(1) \right) \log n\] as \(k,n \to \infty\) in such a way that \(\frac{k}{\log n} \to \kappa\). An effective upper bound in the case \(\frac{k}{\log n} \to \infty\) is also given, improving an earlier result by G. V. Fedorov [Dokl. Math. 88, No. 2, 529–531 (2013; Zbl 1286.11151)].

MSC:

11N56 Rate of growth of arithmetic functions
11N37 Asymptotic results on arithmetic functions
11N64 Other results on the distribution of values or the characterization of arithmetic functions
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References:

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