Good day,
let N be a fixed number and consider a sequence
${r_i= \frac{N}{i}} - floor(\frac{N}{i})\equiv frac(\frac{N}{i}),\; i=1, \dots, N$
Having a finite sequence $(r_i)_{i=1}^{i=N}, \; 0 \leq r_i < 1$ one can fill a histogram with an appropriate bin width, where on the $x$ axis one has $r_i$ and on the $y$ axis one has the number of entries (number of occurences of $r_i$). Once the histogram is filled, one applies some appropriate overall normalization, let me be concrete and chose: the bin containing $r_i=0.5$ should have value one (for example). With $N \to \infty$ the nuber of entries grows and bin size goes to zero. The histogram may possibly become a smooth function. Clearly, I am trying to construct some "$r_i$ density function".
Question is simple: does the limit $N \to \infty$ exist? If yes, which function describes the histogram (in this limit).
SOLVED:
I came to this solution: any sequence of rations $\frac{N}{i}$ for $i=1,\dots,N$ can be divided by $N$ (numerator and denominator) so that N becomes one. If N is big, then (I believe) the sequence $\frac{N}{i}$ can be approximatex by the function $\frac{1}{x}$ where $x \epsilon <0,1)$. One should correctly understand what I mean: numbers $x$ from the interval $<0,1)$ are transofmed by this function. One may think of these numbers as coming from a uniform distribution. Therefore one needs the ask the question: how are numbers from $<0,1)$ transformed by 1/x? The answer is (think about transofming an uniform distribution) 1/x^2, now $x \epsilon <1,\infty)$. In other words, $r_i$ have the same distribution as $\frac{1}{x_i}$ where one can think about $x_i$ as coming from uniform distribution (infitnite statistics). We are almost done: the fractional part intoduces a classes of equivalence: an $x_i$ which ends up as $1.4$ is the "same" (enters the same histogram bin) as a different $x_i$ which ends up as 7.4. This "stratification" simply means that one should, in the resulting $\frac{1}{x^2}$ distribution, add differents intervals of unit length (add their function values) $(1,2> \cup (2,3> \cup (3,4> \dots$ Which translates:
$ f(x)=\sum_{i=1}^{\infty} \frac{1}{(x+i)^2} = \text{HurwitzZeta}(2,x+1) $
Which is the final answer. Special values are $f(0)=\zeta(2)=\frac{\pi^2}{6} = 1.645\dots$ and $f(1)=f(0)-1$.
Picture looks like (SciLab)
