Approximating a Sum (noch nicht ├╝bersetzt)

Problem 756

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, $$\displaystyle S=f(1)+f(2)+f(3)+\cdots+f(n)=\sum_{k=1}^n f(k)$$

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 $0=X_0 < X_1 < X_2 < \cdots < X_m \leq n$ and calculate a modified sum $S^*$ as follows. $$\displaystyle S^* = \sum_{i=1}^m f(X_i)(X_i-X_{i-1})$$

We now define the error of this approximation to be $\Delta=S-S^*$.

Let $\mathbb{E}(\Delta|f(k),n,m)$ 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, $\mathbb{E}(\Delta|k,100,50) = 2525/1326 \approx 1.904223$ and $\mathbb{E}(\Delta|\varphi(k),10^4,10^2)\approx 5842.849907$, where $\varphi(k)$ is Euler's totient function.

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