Consider a function $f(k)$ defined for all positive integers $k>0$. Let $S$ be the sum of the first $n$ values of $f$. That is, $$S=f(1)+f(2)+f(3)+\cdots +f(n)=\sum _{k=1}^{n}f(k).$$

In this problem, we employ randomness to approximate this sum. That is, we choose a random, uniformly distributed, $m$-tuple of positive integers $(X_1,X_2,X_3,\cdots ,X_m)$ such that $0=X_0<X_1<X_2<\cdots <X_m\le n$ and calculate a modified sum $S^{\ast }$ as follows. $$S^{\ast }=\sum _{i=1}^{m}f(X_i)(X_i-X_{i-1})$$

We now define the error of this approximation to be $\mathrm{\Delta }=S-S^{\ast }$.

Let $\mathbb{E}(\mathrm{\Delta }|f(k),n,m)$ be the expected value of the error given the function $f(k)$, the number of terms $n$ in the sum and the length of random sample $m$.

For example, $\mathbb{E}(\mathrm{\Delta }|k,100,50)=2525/1326\approx 1.904223$ and $\mathbb{E}(\mathrm{\Delta }|\phi (k),{10}^4,{10}^2)\approx 5842.849907$, where $\phi (k)$ is Euler's totient function.

Find $\mathbb{E}(\mathrm{\Delta }|\phi (k),12345678,12345)$ rounded to six places after the decimal point.