A Conjecture Relating Modulo Arithmetic and the Riemann Zeta Function.

Question

I recently created a function that has perplexed many of my fellow amateur mathematicians. It goes something like this: $$f\left(g(x)\right)=\frac{1}{N^{2}}\sum_{n=1}^{N}\left(Ng(x)\operatorname{mod}n\right)$$ $g(x)$ can be any smooth, nonzero function, $N$ any a natural number, and $n$ the index of summation. For its simplicity, I chose $g(x)=e^{-x^{2}}$, a Gaussian function. Here is the graph of $f$ when $N=1$ (red), $N=10$ (green), and $N=100$ (blue):

This implies that in the limit as $N\rightarrow \infty$ of $f$, $f$ might converge to a smooth, limiting function. After some trial and error, I found that this limiting curve might in fact be $$e^{-x^{2}}-\frac{\pi^{2}}{12}e^{-2x^{2}}$$

The constant $\pi^{2}\div{12}$ interests me, because twice this value was proven by Euler to be the infinite sum of inverse squares. So our limiting curve can be rewritten: $$e^{-x^{2}}-\sum_{n=1}^{\infty}\left(\frac{e^{-2x^{2}}}{2n^{2}}\right)=e^{-x^{2}}-\zeta(2)\frac{e^{-2x^{2}}}{2}$$ Where $\zeta(2)$ is the Riemann zeta function evaluated at 2. And this is where I do not have the expertise to go any further. I know that there are relationships between the modulo operation and zeta functions (and their corresponding L-functions), but I want to prove these two statements:

1. That the function $f$ does indeed approach a smooth, nonzero limiting curve in the limit of $N\rightarrow \infty$ (specifically for the Gaussian version).
2. The relationship between the Riemann zeta function evaluated at two and the limit of my modular function $f$ for the Gaussian function.

If anyone has any advice or knowledge to impart on this subject, or wishes to offer full or partial proofs of either of these statements, I would be very thankful.

Answer 1

If $0 < r \leq 1$, then $$\begin{align*} \frac1{N^2}\sum_{n=1}^Nn \lfloor rN/n\rfloor &= \frac1{N^2}\sum_{k=1}^{\lfloor rN\rfloor}\sum_{n=1}^{\lfloor rN/k\rfloor}n \\ &= \frac1{2N^2}\sum_{k=1}^{\lfloor rN\rfloor} \lfloor \tfrac{rN}k\rfloor (\lfloor \tfrac{rN}k\rfloor +1) \\ &\sim \frac12\sum_{k=1}^{\lfloor rN\rfloor} \frac{r}{k} \left(\frac{r}{k} + \frac1N\right) \\ &\to \frac{r^2}2\zeta(2) \end{align*}$$

In particular, this verifies your expression for $g(x) = e^{-x^2}$ and gives an affirmative answer to your first question when $0 \leq g \leq 1$. Specifically, for such functions $$f(g(x)) \xrightarrow{N\to\infty} g(x) - \frac{\zeta(2)}2(g(x))^2.$$