Explicit bound of the divisor function
The author proves that the sum of values of Piltz divisor function up to \(x\) satisfies the upper bound \((x(\log x + C_k)^{k-1})/(k-1)!\) for \(1\le x<3\) and \(k\ge 3\) or \(x\ge 3\) and \(k\ge 2\) , where \(C_k\) is approximately \(0.807k -1\). This improves the famous old upper bound by \textit{K. Mardzhanishvili} [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 22, 387--389 (1939; Zbl 0021.20802)], where in place of \(C_k\) appears \(k-1\). The interesting elementary method is based on mathematical induction. As application, one can obtain improved explicit upper bounds for the class number of number fields.
scientific article; zbMATH DE number 7344691