In number theory, a multiplicative function is a function defined on positive integers such that f(ab)=f(a)f(b) for a,b coprime. E.g. Euler's totient function, sum of divisors and number of divisors are multiplicative functions.

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Euler's Totient function $\forall n\ge3$, if $(\frac{\varphi(n)}{2}+1)\ \mid\ n\ $ then $\frac{\varphi(n)}{2}+1$ is prime

While I was studying Euler's Totient function, $\varphi(n)$, I stumbled upon the marvelous book "Index to Mathematical Problems, 1980-1984" By Stanley Rabinowitz. In this page of the book (link to ...
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10answers
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What's the proof that the Euler totient function is multiplicative?

That is, why is $\varphi (A\cdot B)=\varphi (A)\cdot \varphi (B)$, if A and B are coprime? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its ...
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626 views

Very elementary proof of that Euler's totient function is multiplicative

Well, I know two or three proofs of this fact $$\gcd(m,n)=1\implies \varphi(mn)=\varphi(m)\varphi(n)$$ where $\varphi$ is the totient function. My problem is this: I'd like to explain this to some ...
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1answer
407 views

A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The ...
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Identity in Number Theory Paper

In this paper by Jerry Hu, he defines the function $$f_{s,k,i}\left(u\right)=\prod_{p\mid u} \left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose ...
7
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3answers
106 views

For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$

For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$ My try : Left hand side : $\begin{align} \sum_{d|p^k}\sigma (d) &= \sigma(p^0) + \sigma(p^1) + ...
6
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6answers
770 views

Why doesn't $255 \times 255 \times 255 = 16777215$

Ok, I obviously understand basic multiplication and understand why those don't equal. But in web colors, therr is FFFFFF hexadecimal different colors (or rather $16,777,215$ in base $10$). This amount ...
6
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2answers
188 views

Sum of $n \sigma(n)$

What is known about the asymptotic behavior of $$ -\frac{\pi^2}{18}x^3+\sum_{n\le x}n\sigma(n) ? $$ It seems to be $O(x^{2+\varepsilon})$ but I cannot prove this.
6
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1answer
86 views

$ g(n)= \sum_{d|n}\frac{\phi(d)}d=?$

how to find: $$f(n)=\sum_{d|n} d \phi(d)=? $$, $$ g(n)= \sum_{d|n}\frac{\phi(d)}d=?$$
6
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1answer
131 views

$\tau(n)\phi(n)\ge n$

how to prove $\forall n \in \Bbb N$ $$\tau(n)\phi(n)\ge n$$ $\tau(n)$ is number of positive divisor of $n$ my efford: if $n=p$ is prime then $\tau(p)=2,\phi(p)=p-1,2p-2\ge p$ but how prove for ...
5
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2answers
117 views

Prove that $\sum \limits_{d|n}(n/d)\sigma(d) = \sum \limits_{d|n}d\tau(d)$

How can I prove: $$\sum \limits_{d|n}(n/d)\sigma(d) = \sum \limits_{d|n}d\tau(d)?$$ Few observations : Left side is a sum function and the right side is a number of divisors function. Both the sides ...
5
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4answers
511 views

Euler's totient function of 18 - phi(18)

I am trying to find the phi(18). Using an online calculator, it says it is 6 but im getting four. The method I am using is by breaking 18 down into primes and then multiplying the phi(primes) ...
5
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2answers
93 views

Prove $\sum_{k\mid n}{\mu(k)d(k)}=(-1)^{\omega{(n)}}$

I have the following exercise. I am supposed to show that for all natural numbers $n$, that the following equality holds $$\sum_{d|n}{\mu{(d)}d(d)}=(-1)^{\omega{(n)}}$$ Where $\mu$ is the Mobius ...
5
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1answer
71 views

How prove this $\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$

show that $$\sum_{t|n}(d(t))^3=\left(\sum_{t|n}d(t)\right)^2$$ where $d(n)$ is the number of positive divisors of $n$. see this have simaler $$1^3+2^3+\cdots+n^3=\left(1+2+\cdots+n\right)^2$$ maybe ...
5
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3answers
853 views

Proving $ \phi(mn)=\phi(m)\phi(n) \frac k{\phi(k)}$

Suppose $m,n \in \Bbb N$, $k$ is product of all prime number such that divide $m,n$ How to prove that: $$ \phi(mn)=\phi(m)\phi(n) \frac k{\phi(k)}$$
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What is known about these arithmetical functions?

Let $n=\prod_p p^{c_p}$, $N\in \mathbb N$ and $$ \alpha_N(n)=\prod_p p^{c_p \bmod N}. $$ The function $\alpha_N$ is multiplicative since $\alpha_N(n)\alpha_N(m)=\alpha_N(nm)$ for co-prime $n$ and $m$ ...
4
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4answers
2k views

How to find the nearest multiple of 16 to my given number n

If I'm given any random $n$ number. What would the algorithm be to find the closest number (that is higher) and a multiple of 16. Example $55$ Closest number would be $64$ Because $16*4=64$ Not ...
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1answer
109 views

Is the totient function $\varphi$ invertible?

As title, is the totient function $\varphi: \mathbb{N} \to \mathbb{N}$ invertible?
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422 views

Is there a recursive formula for Euler's Totient function

I have been looking for a recursive formula for Euler's totient function or Möbius' mu function to use these relations and try to create a generating function for these arithmetic functions.
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1answer
125 views

Writing product $\prod_{i=1}^m (p_i^{n_i}-1)$ as a sum

Suppose I have an integer $N$ with prime decomposition $N=\prod_{i=1}^m p_i^{n_i}$. How can I write $$\prod_{i=1}^m (p_i^{n_i}-1)$$ as a sum that only depends on $N$, and not it's prime ...
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1answer
55 views

Closed formula for a multiple Dirichlet convolution of the 1-function with the identity

For two multiplicative arithmetic functions $f,g$ the Dirichlet convolution is defined by $(f\ast g) (n)=\sum\limits_{ab=n}f(a)g(b)$. Convoluting any arithmetic function with the $1$-function ...
4
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1answer
128 views

Euler's totient and divisors count function relationship when $[(\frac{\varphi(n)}{2}+1)\cdot(\frac{\tau(n)}{2}+1)] = n$

I am studying the Euler's totient function $\varphi(n)$ and the divisors count function, $\tau(n)$, also named $d(n)$, and recently opened a question (link here) about the following condition: ...
3
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2answers
66 views

how to find $n\in \Bbb N$ such that: $\tau(n)$is odd

how to find $n\in \Bbb N$ such that: $\tau(n)$is odd $\tau(n)$ is number of positive divisor of $n$
3
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1answer
209 views

Prove that the Möbius function is multiplicative

I'm studying algebra, and I came across some questions on multiplicative functions (that should be number theory though?). One is: prove that mobius function is multiplicative. But I've not been given ...
3
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76 views

Why f(1)=1 for every multiplicative function f?

If $f$ is a multiplicative function with $f(1)\ne0$, then why is $f(1)$ necessarily equal to $1$?
3
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1answer
772 views

if $f(n)$ is multiplicative prove that $f(n)/n$ is also multiplicative.

The question asks that if $f(n)$ is multiplicative to prove that $f(n)/n$ is also multiplicative. This is what I have: So, $f(n)$ is multiplicative means that if $p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ ...
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145 views

Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.

I want to prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$. It's given that $d \mid n$, so we know that $n = dm$, for some $m \in \mathbb{Z}$. Now, I want to show that ...
3
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1answer
55 views

Convolution identity involving the Möbius function $\sum_{d|n,d>0} |\mu(d)| = 2^{\omega(n)}$

I'm learning about the Möbius Inversion Formula but I'm stuck on an exercise which involves the Möbius function. Let $n\in\mathbb{Z}$ with $n>0$ and let $\omega(n)$ denote the number of distinct ...
3
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0answers
96 views

Show that the phi function is multiplicative $\phi(mn) = \phi(m)\phi(n)$ [duplicate]

Show that the phi function is multiplicative $$\phi(mn) = \phi(m)\phi(n)$$ Any nice way to prove this without using induction ? The textbook proof looks bit awkward to me, so I am trying to see if ...
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When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime?

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime? Note, $f$ is multiplicative and $\sigma(n)>1, \;n>1$. Therefore $f(n)$ is prime only when $n=p^\alpha$, with $p$ prime, $\alpha\geq1$. ...
2
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357 views

$\sum_{d|n}(-1)^{\frac nd}\phi(d)={}$?

how to find $$\sum_{d|n}(-1)^{\frac nd}\phi(d)=?$$
2
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1answer
72 views

Let rad(n) = $\Pi_{primes, p|n}$ p.

Let $\operatorname{rad}(n) = \displaystyle\prod_{\stackrel{p|n}{p \text{ prime }}}p$ . I have proven that $\operatorname{rad}(n)$ is a multiplicative arithmetic function. I have also proven that ...
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111 views

The multiplicative property of cofactor matrices

In this question, we just consider square matrices. The cofactor matrix $\mathrm{C}(\mathrm{A}) = (c_{ij})$ of a $n$-by-$n$ matrix $\mathrm{A} = (a_{ij})$ is a $n$-by-$n$ matrix ($n > 0$) with ...
2
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2answers
71 views

Orthogonality de Möbius

Does anyone know how prove that $$\sum_{n\leqslant x}\mu(n)\xi(n) =o(x)$$ when $\xi(n)$ is a multiplicative functions? I found one commentary that exist a connection of this problem with the Theory of ...
2
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2answers
78 views

Proof to a property of Euler's totient function

The property is $$\sum_{d|n}\phi(d) = n$$ And the proof provided is If $d$ divides $n$, let $C_d$ be the unique subgroup of $\mathbb{Z}/n\mathbb{Z}$ of order $d$, and let $\Phi_d$ be the set of ...
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1answer
42 views

Möbius function on a finite poset (X, $\leq$)

I'm having some difficulties with the following problem: Give an example of a finite poset $(X, \leq)$ and elements $a,b \in$ X such that $\mu(a,b)=-4$ where $\mu$ is the Möbius function of $(X, ...
2
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1answer
26 views

Multiplying fractions with an x value

$\left(\sqrt{4+\frac{1}{x}}-2 \right) \cdot \left(\sqrt{4+\frac{1}{x}}+2\right)$ I get $\large\frac{1}{x}$ because the square roots go away and the $2$s multiply to make $-4$, so it's: $4 + ...
2
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1answer
137 views

suppose $\omega(n)$ denote the number of distinct prime factors of n

Suppose $\omega(n)$ denote the number of distinct prime factors of n. Prove that$$|\mu(n)|=\sum_{d|n}\mu(d)*2^{\omega(n/d)}$$ Can any one give me some hints about this problem? Is $\mu(n)$ a ...
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215 views

Group-theoretic proof that an increasing multiplicative function is exponential

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing function such that $$f(2)=2$$ and $$f(mn)=f(m)f(n)$$ for all positive integers $m,n$ such that $$\gcd(m,n)=1$$ Prove that ...
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1answer
58 views

Proving that $f$ is differentiable at $0$

Let's consider the following function: $$f(x,y)=\begin{cases} (x^2+y^2)\sin\left(\dfrac{1}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$ I know that ...
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1answer
33 views

Number theoretic function related to totient

I'm doing an excercise in Alan Baker's book A Concise Introduction to the Theory of Numbers, and I'm confused about the method spelled out for one question. I'll quote it here: Let $a$ run through ...
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1answer
31 views

Prove that $f$ is a multiplicative function and calculate the Summatory Function

Define $f(n)$ as $1$ if $n$ is odd, and $3$ if $n$ is even. So i have $f(odd) = 1$ and $f(even) = 3$ If a function $f$ is multiplicative then if $gcd(m,n) = 1$ then $f(m * n) = f(m) * f(n)$ This ...
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1answer
49 views

Summation of multiplicative function $f$ where $f(p) = 1$ for $p$ prime

I have a multiplicative function $f$ with a special "base" case: For every prime $p$, $f(p) = 1$. E.g. splitting up $f(3^5 \times7^2 \times 13 \times 17)$ yields $f(3^5) f(7^2)$ which is left to be ...
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4answers
51 views

How to find all elements in Z/80 that have multiplicative inverses.

I need to find all the elements in Z/80 that have multiplicative inverses. Z/80 is not a field, so I know not every element will have an inverse. Is there a shorter way than just writing out the ...
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1answer
111 views

Proof that the euler totient function is multiplicative, correctness?

I've tried proving that $\varphi(mn) = \varphi(m)\varphi(n)$ (if $gcd(mn)=1$). The proof I try to setup doesn't look like the proof I find in textbooks, where am I going wrong? Proof: We try to ...
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1answer
80 views

calculating storage based upon on eps?

If an organization collects an average of $20,000$ EPS over eight hours of an ongoing incident, that will require sorting and analysis of $576,000,000$ data records. Using a $300$ byte average size, ...
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1answer
23 views

$a\cdot(b^{-1}\bmod m)$ Can be be solved using modular multiplication

Does $a\cdot(b^{-1}\bmod m) = (a\bmod m) \cdot(b^{-1}\bmod m).$ where $\bmod$ represents remainder left on division with $m$. $b^{-1} \bmod m$ is multiplicative inverse.
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1answer
75 views

Prove $\lambda(n)=\sum_{d^2|n}\mu(n/d)^2$ and $\mu^2(n)=\sum_{d^2|n}\mu(d)$

$\lambda(n)$= $\sum_{d^2|n}$ $\mu(n/d)^2$ and $\mu^2(n)$= $\sum_{d^2|n}$ $\mu(d)$ Having a little bit of trouble here.Can I use the fact that $\sum_{d|n}\lambda(n)$ is a characteristic function for ...
1
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0answers
23 views

An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
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0answers
358 views

Modular multiplicative inverse and coprime numbers needed.

I have a 64 bit algorithm that uses modular multiplicative inverse and coprime numbers, and I need to convert it to 32 bit. This math is not my area, and I cannot find an online calculator, so I hope ...