All Questions
27
questions
2
votes
1
answer
200
views
A divisor sum involving Moebius function and Jordan's totient function
I am trying to prove the following claim:
Let $\mu(n)$ be the Moebius function and let $J_k(n)$ be the Jordan's totient function. Then,
$$\displaystyle\sum_{d \mid n} \frac{\mu^2(d)}{J(k,d)}=\frac{n^...
- 12.9k
3
votes
1
answer
424
views
Calculating a sum with Euler's totient function
If $S(a,b)= \{ k \in \mathbb{Z} \mid a ~ (\mathrm{mod} ~ k) + b ~ (\mathrm{mod} ~ k) \ge k \}$, calculate the following sum
$$\sum_{k\in S(m,n)}\varphi(k)$$
where $\varphi$ is Euler's totient function....
user775075
2
votes
1
answer
127
views
Proving $\sum d\mu \left(\frac{n}{d}\right) = \frac{\mu\left(\frac{n}{(m,n)}\right)}{\phi \left(\frac{n}{(m, n)}\right)}\phi(n)$.
I came across the formula $\sum_{d|\text{gcd}(m,n)} d\mu \left(\frac{n}{d}\right) = \frac{1}{\phi \left(\frac{n}{\text{gcd}(m, n)}\right)}\mu\left(\frac{n}{\text{gcd}(m,n)}\right)\phi(n)$, where $m,n \...
- 21
4
votes
2
answers
180
views
Phi Function Summation and Divisors
Suppose $n$ is a composite positive integer. Is there a simple way to express the sum
$$
\sum _{d|n} \frac{\varphi(d)}{d}?
$$
what about
$$
\sum_{d|n} d \varphi(d)?
$$
I know that the sum $\sum _{d|...
- 525
1
vote
1
answer
146
views
Estimating the sum reciprocal of the Totient Function
I'm going through "Additive Number Theory: The Classical Bases" at the moment, and I'm having difficulty with a certain set of inequalities. Namely,
$$\sum_{q>Q}\frac{1}{\phi(q)^2} \leq \sum_{q>...
- 221
2
votes
1
answer
285
views
Sum of quotient of Euler phi functions
Let $n$ divide $c$. I would like to find a closed form for the expression
$$\sum_{k \mid c} \frac{\varphi(kn)}{\varphi(k)},$$
where $\varphi$ is the Euler phi function. Because $n$ divides $c$, it is ...
- 153
0
votes
1
answer
55
views
Looking for the name and properties of ${\varphi}_{2} (r, N) = \sum_{- N \le s, t \le N, (r, s, t) = 1} 1$ and $\sum_{d \mid r} \mu (d)/d^2$
I am counting the number of unique polynomial candidates for a fixed $r$ where $1 \le r \le N$ with $|s|, |t| \le N$ for naive height $N \ge r$. This sum is $${T}_{2} \left({r, N}\right) = \sum_{\...
- 967
3
votes
2
answers
367
views
Sums related to the Euler totient function
I'm trying to estimate asymptotically the following sums:
$$
S_1(m, n) = \sum_{1\leq i \leq n, (m,i)=1}{\frac{1}{i}}
$$
$$
S_2(n) = \sum_{i=1}^n{i\phi(i)},
$$
where $(m,i) = GCD(m,i)$, and $\phi(i)$ ...
- 43
2
votes
2
answers
294
views
Summing the Euler $\varphi$-function $\varphi(n)$
The Euler $\varphi$-function $\varphi(n)$ counts the number of positive integers less than or equal to $n$,
which are relatively prime with $n$.
I would like to evaluate
$$ \sum_{d\mid n}\varphi(d) $...
- 33
5
votes
2
answers
2k
views
Relation between $\gcd$ and Euler's totient function .
How to show that $$\gcd(a,b)=\sum_{k\mid a\text{ and }k\mid b}\varphi(k).$$
$\varphi$ is the Euler's totient function.
I was trying to prove the number of homomorphisms from a cyclic group of order $...
- 2,646
3
votes
3
answers
2k
views
Clarification on proof of the sum of Euler $\phi$ fcn: $\sum_{d|n}\phi(d)=n$
In the proof, it says "clearly equals $\phi(n_1)$". I don't see how this is clear. I also don't see how this implies $n=\sum_{d|n}\phi(n/d)$.
Can someone please clarify this proof?
(from A Course in ...
- 12.1k
-1
votes
1
answer
69
views
Dealing with phi function property
If $n=2^kN$, where $N$ is odd, then
$$\sum_{d\mid n}(-1)^{n/d}\phi(d)=\sum_{d\mid 2^{k-1}N}\phi(d)-\sum_{d\mid N}\phi(2^kd)$$
I have no idea how to seperate things inside the left side. In a nornal ...
- 785
2
votes
1
answer
707
views
Euler's totient function and related sum
I am supposed to calculate the following sum :
$\sum _{d|n} (n/d) * \phi (n/d)$ where the sum is over all divisors (d) of a given number (n) and $\phi (x)$ is Euler's totient function .
Since the ...
user180000
3
votes
2
answers
2k
views
Connection between GCD and totient function
I found the following formula which connects Euler's totient function with gcd at wikipedia.
$$ \gcd(a,b) = \sum_{k|a \; \hbox{and} \; k|b} \varphi(k). $$
The problem is that I can not figure out ...
- 2,665
-1
votes
2
answers
1k
views
How to show that $ \sum_{d/n} \mu^{2}(d)/\phi(d) = n/\phi(n)$? [closed]
$\forall n, n\in\mathbb{N}$ $\frac{n}{\phi{(n)}} = \sum_{d/n} \frac{\mu^{2}(d)}{\phi(d)}$
Where $\mu$ is the Möbius function.
