Linked Questions

2
votes
2answers
286 views

For all $X \geq 1$,$\sum_{1 \leq n \leq X} \mu(n) \left[\frac{X}{n}\right] = 1.$ [duplicate]

I'm currently sitting with the following number theory problem: Prove that for all $X \geq 1$, $$\sum_{1 \leq n \leq X} \mu(n) \left[\frac{X}{n}\right] = 1.$$ A few ideas I have tried: Using that $[...
4
votes
1answer
310 views

Prove $\sum_{k = 1}^n \mu(k)\left[ \frac nk \right] = 1$ [duplicate]

I need to prove the identity $$\sum_{k = 1}^n \mu(k)\left[ \frac{n}{k} \right] = 1$$ where $n$ is a natural number, and $[n]$ denotes the floor function. The proof also should not use the Möbius ...
43
votes
4answers
3k views

Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$. Challenge: Prove $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\...
4
votes
1answer
333 views

$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$ Identity involving Liouville Lambda function

I have to prove $$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$$ I tried using the approach in this question but I don't know how I'll get $\sqrt{N}$. Please help.
7
votes
1answer
193 views

What is the inverse of $\left[ \sum_{k=1}^{j} \left\lfloor \frac{i}{k} \right\rfloor \right]_{n \times n}$?

For $n \in \mathbb{N}$, let $M_{n}$ denote the $n \times n$ integer matrix whereby the $(i, j)$-entry of $M_{n}$ is equal to $\sum_{k=1}^{j} \left\lfloor \frac{i}{k} \right\rfloor$, for all indices $i$...
3
votes
1answer
494 views

An application of Mobius Inversion $\sum_{d \mid n} \mu(\frac{n}{d})\nu(d) = 1$

Show that $\sum_{d \mid n} \mu(\frac{n}{d})\nu(d) = 1$, for any positive integer n. Where $\mu$ denotes the Mobius function defined by $\mu(n)=(-1)^{s}$ if $n=p_{1} \dotsc p_{s}$ for distinct primes $...
2
votes
1answer
342 views

Restating a floor function as a finite sum

It seems to me that a floor function can be expressed as a finite sum that is open to the Möbius function. Does it follow for all nonnegative integers $a$ that: $$\left\lfloor\frac{a}{b}\right\...
2
votes
2answers
107 views

Prove $ \left\vert \sum_{n=1}^N \frac{\mu(n)}{n} \right\vert \leqslant 1,$ where $\mu(n)$ is the Mobius function.

Given a positive integer $N$, show that $$ \left\vert \sum_{n=1}^N \frac{\mu(n)}{n} \right\vert \leqslant 1,$$ where $\mu(n)$ is the Mobius function. How do I approach this question? I guess a ...
1
vote
2answers
91 views

Alternate approach to show $|\sum_{i \leq n} \frac{\mu(i)}{i}| \leq 1$?

I am doing a question out of H. E. Rose's A Course In Number Theory (Chapter 2, problem 2) which I have been struggling with for some time. However I found a solution which strays from the way advised ...
2
votes
1answer
48 views

A proof of the approximate expression about totient summatory function

I'm a high school student in Korea. I am preparing for a presentation. so I prove an approximate expression about totient summatory function , but I'm not sure that the proof is correct. If the ...