Irrational factor of order \(k\) and its connections with \(k\)-free integers
If \(n=\prod_{i=1}^\ell p_i^{\alpha_i}\) is the prime fatorisation of \(n\), define the irrational factor \(I_k(n) = \prod_{i=1}^\ell p_i^{\beta_i}\) where \(\beta_i=\alpha_i\) if \(\alpha_i<k\) and \(\beta_i=1/\alpha_i\) if \(\alpha_i\geq k\). The idea is that \(I_k(n)\) measures how far \(n\) is from being \(k\)-power free. More precisely, if \(\mu_k(n)\) is the characteristic function of the \(k\)-free integers, then \[ \bigg| \frac{I_k(n)}{n} - \mu_k(n)\bigg| \leq \frac{1}{2^{k-\frac{1}{k}}}, \] so that \(I_k(n)/n\) is very close to \(\mu_k(n)\) for large \(k\). In fact, the difference is 0 if \(n\) is \(k\)-power free. The main results of the paper give asymptotic formulae for the geometric mean and the arithmetic mean of \(I_k(n)\) by using the machinery of analytic number theory (estimates from the distribution of primes and zero-free regions for the \(\zeta\)-function and Perron's formula). A simple consequence is that the average of \(I_k(n)/n\) is \(K_k(2)/\zeta(k)\) where \(K_k(2)\) is a certain constant which depends ony on \(k\) and \(\lim_{k\rightarrow\infty} K_k(2) = 1\).
scientific article; zbMATH DE number 6384130