My Take On Champernowne's Constant

\[ \large 0.12345678910111213\ldots \]

The number above is an irrational decimal number created by concatenating all the positive integers in ascending order, and it is called Champernowne's constant.

Let the \(i^\text{th}\) digit of fractional part of Champernowne's constant (base 10) be denoted by \(a_{i}\). Calculate the digital sum of product of all non-zero \(a_{i}\) where \(i\) is a prime number less than 1000.

Clarification:

The digital sum is sum of all digits of a number. For example, product of \(a_i\) where \(i\) is prime less than \(8\) is \(2\times 3\times 5\times 7=210\). Its digital sum is \(2+1+0=3\).

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