Totient Sum

Let $S (n, m) = \sum ϕ (n \times i)$ for $1 \leq i \leq m$ . ( $ϕ$ is Euler's totient function)
You are given that $S (510510, 10^{6}) = 45480596821125120$ .

Find $S (510510, 10^{11})$ .
Give the last $9$ digits of your answer.

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.