I get from an artificious way, but simple, two similar statements that I belive that are true (I had that identify some arithmetic functions), that is my Claim below. In next question I am asking about how to create good conjectures concerning two erratic arithmetic function, that will be introduced in this statement (see my motivation).
Motivation. M1) For $n\geq 1$ was defined in the literature the sum of remainders function $$s(n)=\sum_{k=1}^n n\operatorname{mod}k.$$ Professor Spivey did a comparison in [1] between the properties that the sum of divisor function has versus this function. Is known that there are interesting identities with weights involving functions like Pillai function (I try to explore different ways to know more about the sum of remainders function).
M2) From simple identities that appears in [2] multiplying by, corresponding the Möbius function $\mu(n)$ or the Liouville function $\lambda(n)$, and taking the sum, I can state the following (notice of the typography $\nmid$ in the first terms of $LHS$)
Claim. A) For each $n\geq 1$ $$\varphi^{-1}(n)+\sum_{\substack{1\leq k\leq n \\ k\nmid n}}\mu(k)(n\operatorname{mod}k)=M(n)+\sum_{1\leq k\leq n}\mu(k)((n-1)\operatorname{mod}k)$$ where $\varphi^{-1}(n)$ is the Dirichlet inverse of Euler's totient function, and $M(n)$ is the Mertens function.
B) For each $n\geq 1$ $$f(n)+\sum_{\substack{1\leq k\leq n \\ k\nmid n}}\lambda(k)(n\operatorname{mod}k)=L(n)+\sum_{1\leq k\leq n}\lambda(k)((n-1)\operatorname{mod}k)$$ where $f(n)$ is the Sloane's sequence A061020 (see [3] for the rest of references of our sequences), and $L(n)$ is the summary function $$L(n)=\sum_{k=1}^n\lambda(k).$$
M3) I have curiosity also about how to create good conjectures involving erratic functions. I am asking about how to combine mathematical reasonings, knowledges of the behaviour of the partial sums of Möbius and Liouville functions, and/or experiments with a computer to set conjectures that would be difficult to rule out.$\square$
Question. I was doing experiments and seems that the arithmetic functions defined for integers $n\geq 1$ $$R_1(n):=\sum_{1\leq k\leq n}\mu(k)((n-1)\operatorname{mod}k),$$ and $$R_2(n):=\sum_{1\leq k\leq n}\lambda(k)((n-1)\operatorname{mod}k)$$ are very erratic*. Imagine that I have a friend that ask me about a reasoning/method to get good conjectures about the asymptotic behaviour of these functions as $n\to\infty$. Are possible deductions to set such conjectures using mathematical ideas (reasonings, heuristics, numerical evidence)? What should be such conjectures? Many thanks.
*You can see the behaviour of the sequence typying similar codes than these in Wolfram Alpha online calculator:
sum mu(k)mod(10000-1,k), from k=1 to 10000
sum LiouvilleLambda(k)mod(50000-1,k), from k=1 to 50000
References:
[1] Spivey, The Humble Sum of Remainders Function, Mathematics Magazine Vol. 78, No. 4 (2005).
[2] James T. Cross, A Note on Almost Perfect Numbers, Mathematics Magazine, Vol. 47, No. 4 (1974).
[3] The sequences A002321, A002819 and A023900 from The On-Line Encyclopedia of Integer Sequences.
Let $h(n) = \sum_{k=1}^n \mu(k) (n \bmod k)$.
Note that $(n \bmod k) - (n-1 \bmod k) = 1-k\, 1_{k |n}$ thus
$$h(n)-h(n-1) = \mu(n)(n \bmod n) +\sum_{k=1}^{n-1} \mu(k)(1-k\, 1_{k |n}) = M(n)- (f(n)-\mu(n)n)$$
where $f(n) = \sum_{k | n} k\mu(k) $ is the Dirichlet inverse of $\varphi(n)=\sum_{d | n} \mu(d) \frac{n}{d}$.
So understanding $f(n)$ and $M(n)=\sum_{k=1}^n \mu(k)$ is enough for understanding $h(n)$, and that's what you should have seen before posting this.