Denoting as

$$s(n)=\sum\_{k=1}^n n \bmod k$$ **the sum of remainders function** (each remainder is defined as in the euclidean division of integers $n\geq 1$ and $k$). See [1] for example.

**For examples** $s(1)=0$ and
 $$s(6)=0+6\bmod4+6\bmod5+0=2+1=3.$$

After some **computational experiments** I am interesting in

> 
> **Question.** Can you prove or refute (too are welcome heuristics) that $\forall\epsilon>0$ $$\lim\_{n\to\infty}\frac{s(n-1)}{\epsilon+s(n)}=1?$$
>  Similarly that $\forall\epsilon>0$ $$\lim\_{n\to\infty}\frac{s(n)}{\epsilon+s(n-1)}=1?$$
>  **Thanks in advance**, it seems curious, and I believe that previous is a strong condition in the sense that its proof could be easy, or well in the sense that find a counterexample could be easy (I am refering to the presence of such $\epsilon$).
> 
> 
> 

**I don't know if previous exercises are in the literature.**

There are formulas involving this function (see [1] for example) with the **sum of divisors function** $\sigma(n)=\sum\_{d\mid n}d$, that holds for each integer $n>1$

$$\sigma(n)+s(n)=s(n-1)+2n-1,$$ 

and **a companion formula** that can be deduced inductively from previous (see too in [1]).

Well my experiements aren't the bests, neither I have enough *resolution to see the limit*, but **there is a difference** when I plot $\frac{s(n)}{s(n-1)}$ ($n\geq 5$) or $\frac{\sigma(n)}{\sigma(n-1)}$ or $\frac{\phi(n)}{\phi(n-1)}$, where this last is Euler's totient function. 

**References:**

[1] Math Overflow, How to calculate the sum of remainders of N? <https://mathoverflow.net/questions/195325/how-to-calculate-the-sum-of-remainders-of-n>

I've read the notes from two authors Spivey and Cross.