IL-GILLIS 2017 Problem 2

Denote by $P(n)$ the product of the digits of a positive integer $n$ . For example, $P(1948)=1\cdot9\cdot4\cdot8=288$ .

[list=a] [] Evaluate the sum $P(1)+P(2)+\dots+P(2017)$ . [] Determine the maximum value of $\frac{P(n)}{n}$ where $2017\leq n\leq5777$ . [/list]