Large values and champion numbers of Kalmár's arithmetic function
Kalmár's arithmetic function \(K(n)\) counts the number of ordered factorizations \(x_1\cdots x_r\) of \(n\) in integer parts \(x_i\geq 2\) for \(i=1,\dots,r\). In the paper under review, the authors study the champions of this function, which are the positive integers \(n\) such that \(K(n)>K(m)\) for all \(m<n\). In 1931, \textit{L. Kalmár} [Mat. Fyz. Lapok 38, 1--15 (1931; Zbl 0003.24604; JFM 57.1366.01)] showed that the summatory function \(\sum_{n\leq x} K(n)\) of \(K(n)\) is asymptotically \(c_1x^{\rho}\), where \(c_1\) is some positive constant and \(\rho\) is the only real zero of \(\zeta(\rho)=2\). In the paper under review, it is shown that there exist two positive constants \(c_2\) and \(c_3\) such that inequality \[ \log K(n)\leq \rho \log n-c_2{{(\log n)^{1/\rho}}\over{\log\log n}} \] holds for all sufficiently large \(n\), whereas the inequality \[ \log K(n)\geq \rho \log n-c_3{{(\log n)^{1/\rho}}\over {\log\log n}} \] holds for infinitely many positive integers \(n\). This sharpens a result of \textit{M. Klazar} and the reviewer [J. Number Theory 124, No. 2, 470--490 (2007; Zbl 1169.11043)]. Regarding champions for the function \(K(n)\), it is shown that there exist two constants \(c_4\) and \(c_5\) such that if \(N\) is champion for \(K\), then its total number of prime factors is asymptotically \(c_4\log N\) as \(N\to\infty\), whereas the number of distinct prime factors of \(N\) is asymptotically \(c_5(\log N)^{1/\rho}/(\log\log N)\) as \(N\to\infty\). There are also upper and lower bounds for the counting function of the number of champions up to some large real number \(X\). The paper concludes with several open problems which are likely to spur further research on the topic.
scientific article; zbMATH DE number 5285740
scientific article