Proving an Inequality Involving the Summation of Modulo Functions

Question

I have a function, $c(n)$, so that $$c(n)=\sum_{i=1}^n n \bmod i$$ I also have the function $f(n)$ and $g(n)$, so $$f(n)=\left(c\left(n-1\right)+n-1\right)-c\left(n\right)+n$$ and $$g(n)=e^{\gamma}n\ln\ln n$$ with $\gamma$ being the Euler-Mascheroni constant. Is it possible to prove that for large enough $n$, $f(n)<g(n)$?