# Set conjectures concerning the asymptotic behaviour of erratic arithmetic functions, related to the Möbius function and the Liouville function

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.

• What is your question ???? You need to stop with those kind of posts and just wait until you can find a 4 line statement with a well-defined question. May 19 '17 at 17:29
• Many thanks @user1952009 for your attention and answer, I belive that previous question is well-defined. You can read it: are possible deductions to set conjectures concerning the asymptotic behaviour of each (some of) $R_i(n)$ as $n\to\infty$? (And if it is possible explain the ideas of how state those conjectures).
– user243301
May 19 '17 at 17:59
• Lis ma réponse. $f(n)$ et $\mu(n) n$ sont des fonctions multiplicatives liées à $\zeta(s)$. L'analyse qu'on peut en faire est la même que pour $M(n)$ et $\mu(n)$ (commence - comme tout le monde - par étudier une démonstration du théorème des nombres premiers..) May 19 '17 at 18:18
• @user1952009 I am agree with your words in french language. And that some of my questions in this site are atypical. My purpose is ask friendly questions, and I hope interesting, with the purpose to learn more mathematics. Well and I accept that some times I could am wrong with my questions. Ok, many thanks any case.
– user243301
May 19 '17 at 18:33

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.