The Mertens function is defined as follows: (1) $\quad\mathcal{M}(N)=\sum_{n=1}^N\mu(n)$ I have a very simple question for which I cannot seem to find a definitive answer on the web. I’ve also consulted several books on number theory and can’t even seem to find a conjecture much less a proven result. Question 1:: Does the Mertens function […]

The Chaos Game is the famous method to create fractals elaborated by professor Michael Barnsley. As Wikipedia explains: “The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one […]

About 4 minutes into his video on chordal graphs, Donald Knuth mentions a variant of the Möbius Function for graphs. I was curious what it meant outside the context of number theory?

I’ve asked this, after I was doing the calculations that I show below Question. Let $\mu(n)$ the Möbius function, and $\pi(n)$ the prime-counting function. How one can to prove that $$\sum_{n=1}^\infty\frac{\mu(n)\pi(n)}{n^2}$$ does converge? The calculations of my motivation were to combine the Prime Number Theorem and this asymptotic formula 27.12.3 from the Digital Library of […]

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where $c_k = e_k – 1$ Example: $72 = 2^33^2$, so $f(72) = 2^{3-1}3^{2-1} = 2^{2}3^{1}=12$ $144 = 2^43^2$, so $f(144) = 2^{4-1}3^{2-1} = 2^{3}3^{1}=24$ Now let $$F(N) […]

Counting square free numbers $\le N$ is a classical problem which can be solved using inclusion-exclusion problem or using Möbius function (http://oeis.org/A071172). I want to count square free numbers which are co-prime to a given number $m$ within a limit. Let $C(N, m)$ = no. of square free numbers $\le N$ and co-prime to $m$. […]

I’m trying to work through Ireland and Rosen’s A Classical Introduction to Modern Number Theory as I’ve heard good things about it. This is Exercise 12 from Chapter 2. Here $\mu$ is the Moebius function, and $\phi$ the totient function. Find formulas for $\sum_{d|n}\mu(d)\phi(d)$, $\sum_{d|n}\mu(d)^2\phi(d)^2$, and $\sum_{d|n}\mu(d)/\phi(d)$. Playing around with the first sum, I know […]

A while ago, I answered this question here on StackExchange which asks if for any given integer $k$, whether there exists infinitely many natural numbers $n$ such that $$ \mu(n+1)=\mu(n+2)=\mu(n+3)=\cdots=\mu(n+k) $$ This is indeed the case, and can be shown using the Chinese Remainder Theorem, as in my answer to the linked question. We find […]

I am trying to show $$\sum_{d \leq x} \mu(d)\left\lfloor \frac{x}{d} \right\rfloor = 1 \;\;\;\; \forall \; x \in \mathbb{R}, \; x \geq 1 $$ I know that the sum over the divisors $d$ of $n$ is zero if $n \neq 1$. So we can rule out integers that are divisors of $x$ (if $x > […]

Suppose that $p$ is a prime. Prove that the sum of the primitive roots modulo $p$ is congruent to $\mu(p − 1) \pmod{p}$.

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