On the number of polynomial maps into \(\mathbb Z_n\) (Q875688)

The authors prove interesting theorems for the maximal, minimal normal and average orders of the function \[ f(n):= \prod_{0\leq k\leq n} {n!\over \gcd(n, k!)}, \] which is the cardinality of the set of polynomial maps from \(\mathbb Z\) into \(\mathbb Z/n\mathbb Z\). Theorem 1. The inequality \[ \log f(n)\geq \{1/2+o(1)\}(\log n)^2/\log\log n \] holds as \(n\to\infty\). Theorem 2. For all but \(O(x(\log\log x)^2/\log x)\) positive integers \(n\leq x\), we have \[ \log f(n)=\{1+O((\log\log x)^2/\log x)\}P(n)\log P(n), \] where \(P(n)\) denotes the largest prime factor of \(n\). Theorem 3. Let \(\nu>0\). Then \[ \sum_{n\leq x} (\log f(n))^\nu = \left\{{\zeta(\nu+1)\over \nu+1} +O\left({(\log\log x)^2\over \log x}\right)\right\} x^{\nu+1}(\log x)^{\nu-1}, \] where \(\zeta\) is the Riemann zeta-function.