The upper limit value of the divisor function with growing dimension (Q393851)

Let \(\tau_k(n)\) be the number of representations of a positive integer \(n\) by product \(n=x_1x_2\cdots x_k\) with positive integers \(x_1,x_2,\ldots, x_k\). In addition, it is assumed that \(\tau_k(0)=0,\tau_k(1)=1\) and \(\tau_1(n)=1\). In the case \(k=2\), the value of \(\tau_2(n)\) equals to the number of different divisors of \(n\). In the paper, the case when \(k\) goes to infinity together with \(n\) is considered. For instance, it is shown that the upper limit \[ \limsup\limits_{n\rightarrow\infty}\frac{\log_2\tau_k(n)\cdot\log_2\log_2n}{\log_2n\cdot\log_2k}=1 \] is attained if \(k=k(n)\rightarrow\infty\) and \(k(n)/\log_2n\rightarrow 0\).