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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Lagarias and Robin theorems versus multiplicative property

If I use for example Robin's theorem, see here in the section Growth of arithmetic functions, or Lagarias equivalence, see (5) here has sense ask us what is the more sharp inequality for ...
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A Increasing Multiplicative Functional Equation where $nm$ is a cube

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 $mn$ is a perfect cube. ...
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Proof of inequality involving multiplicative function?

The identity below seems true for the examples I've considered. I thought I had proven it using induction but found a mistake and removed my attempted proof since it is not helpful. Given: $P(z)$ is ...
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Can't find the error with the following sum involving multiplicative functions

I apologize for this question which might be trivial but I'm stuck with this issue and I'm probably doing some mistake over and over again. Here's the framework. Let $f$ and $h$ be two multiplicative ...
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Hints to compute if exists $\lim_{n\to\infty}\sum_{k=1}^n\sigma(k^2)/\sum_{k=1}^n\sigma(k)$, which $\sigma(n)=\sum_{d\mid n}d$, and other question

I would like receive hints at least for one of the following problems, these are going from experiments. Can you provide to me hints for at least one of the following problems? I will try put the ...
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Counting divisors of a number

Let m be any positive integer and consider $\Sigma_{d|m} \frac{1}{d} $. I wish to ask whether there is a closed form expression for the above sum.
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Average Order of $\frac{1}{\mathrm{rad}(n)}$

Again a question about $\mathrm{rad}(n).$ Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing $n$. Or equivalently, ...
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7answers
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How to compute $8x \equiv 33 \pmod{35}$?

How to compute $8x \equiv 33 \pmod{35}$? I followed this video to solve this problem. Is there a better way? My solution steps: Divide both sides by 8: $$x \equiv \frac{33}{8}^{-1} \pmod{35}$$ ...
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Series of the totient function

Good morning, I wonder if : $$\sum_{n} \frac{(-1)^n}{\varphi (n)}$$ converges or not. where $\varphi (n)$ is the Euler function. Do you have any idea ?
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Number of zeros of a polynomial modulo n is a multiplicative function

Let $f$ be a polynomial with integer coeffcients. For $n\geq1$ let $N_f(n)$ denote the number of pairwise incongruent solutions of $f(x)=0$ mod n. I need help proving that $f$ is a multiplicative ...
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Looking applications for these statements involving multiplicative functions and Euler-Fermat theorem

It is well knwon that for positive integers such that $gcd(a,n)=1$, Euler-Fermat Theorem states than $a^{\phi(n)}\equiv 1 \mod n$, where $\phi(n)$ is the Euler totient function counting positive ...
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Determine if $f(n) = n+k$ is completely multiplicative, multiplicative or neither.

Question Determine if $f(n) = n+k$ (k is a fixed real number) are completely multiplicative, multiplicative or neither. Attempted solution The only background I have at this point are (1) the ...
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A follow-up number-theory question on the deficiency function $D(x) = 2x - \sigma(x)$

This question is a follow-up to these previous posts: MSE1 and MSE2. Let $x, y$ be positive integers. We call $\sigma(x)$ the sum of the divisors of $x$. Let the deficiency function $D(x)$ be ...
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Over what subset of $\mathbb{N}$ is the deficiency $D(x) = 2x - \sigma(x)$ a weakly multiplicative function?

This is an offshoot of this MSE question which was posted earlier today. Let $\mathbb{N}$ be the set of natural numbers (i.e., positive integers). We call $\sigma(x)$ the sum of the divisors of $x$. ...
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1answer
76 views

Composition of polynomial and multiplicative is multiplicative .

I made the following problem a while ago but I can't solve it (also I don't think it's extremely hard ) : Let $f$ be a non-constant completely multiplicative function over $\mathbb{Z}$ . Assume ...
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134 views

Average order of $\mathrm{rad}(n)$

Let $\mathrm{rad}(n)$ denote the radical of an integer $n$, which is the product of the distinct prime numbers dividing n. Or equivalently, $$\mathrm{rad}(n)=\prod_{\scriptstyle p\mid n\atop p\text{ ...
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1answer
74 views

If $\varphi(mn)=\lambda \varphi(m)\varphi(n)$ what should be written for $\lambda$

Respected All. I am studying number theory where I came to know that $\varphi(n), \sigma(n)$ both are multiplicative function ; In other words, if $(m,n)=1$ then \begin{align} ...
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How can I find The Multiplicative Inverse of $1+\sqrt{2}$? [closed]

I am doing contemporary abstract algebra and am working in an integral domain. I have found it necessary to compute the multiplicative inverse of $1+\sqrt{2}$; I know such the definition of a ...
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$a\cdot(b^{-1}\bmod m)$ 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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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 $m$ ...
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131 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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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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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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55 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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44 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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191 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: ...
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51 views

Proving that a summation is multiplicative

I have been give a project for number theory: For $m>0$ , let $f(m) = \sum_{r=1}^m \frac{m}{\gcd(m,r)}$ . Evaluate $f(m)$ in terms of the prime factorization of $m$. So far, I have found a formula ...
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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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If a function $f$ is multiplicative, how do I show that $\sum_{d\mid n} \mu(n/d) f(d)$ is also multiplicative?

I am studying A Classical Introduction to Modern Number Theory by Ireland and Rosen, and this is exercise 9 from chapter 2. The authors define a function $f$ to be multiplicative if for all $a, b$ ...
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formula for the number of perfect squares mod $N$

In a numerical experiment I notice for sum moduli $N$ there are much less than $N/2$ perfect squares. I had chosen a large number, the simplest example is $N=8$. Using the Chinese Remainder Theorem ...
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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 ...
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new addition and new multiplication x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z, prove the set Z equipped with these 2 new operation

here says a new operation addition and new operation multiplication x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z,where the operations on the right hand side are ordinary addition and multiplication ...
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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, ...
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Why is this expression returning NaN? [closed]

For smoother ship movement, I am going to gradually move it to it's actual location - So like this: (mz - sz) / (10 * (60 / TerrainDemo.FPS)) mz is the ship's ...
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381 views

Grade School Multiplication Algorithm for Binary Numbers explanation

I under stand the shifting but not why it will always give the right answer? For Example: ...
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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 ...
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Why is 15 + 15 different from 15 * 2?

(Apologies if the tag is incorrect. I can't find a "Multiplication" tag or similar) I'm going to be adding a set of 5 numbers up and dividing them to get the average for a java game I am making. ...
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sum of divisors function $\sum \tau(n) = \frac{1}{4}$

These notes on multiplicative number theory mention the convolution $ 1 \ast 1 = \tau$ (where $\tau$ is the divisor function not Ramanujan tau function. Therefore $$ \bigg(\sum \frac{1}{n^s} ...
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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 ...
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Prove that $g(n)$ is a multiplicative function as well

Suppose that $f(n)$ is any multiplicative function , and define a new function as $g(n) = f(d_1) + f(d_2) +...+f(d_r)$, where $d_1, d_2,...,d_r$ are divisors of $n$. Prove that $g(n)$ is a ...
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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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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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Simple act practice test problem

I am going through an act practice test and I came to a problem that said. Which of the following is equal to the product of x and the square of its reciprocal for all x < 0. My first step ...
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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 + ...
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Euler-Totient Multiplicative

http://www.oxfordmathcenter.com/drupal7/node/172 By and large, I understand this proof, however I'm struggling to understand how the Chinese remainder theorem implies that there exists some $x \in ...
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How to show that

If $f$ is a multiplicative function proof that: i) $f^{-1}(p^{2})= [f(p)]^{2}-f(p^{2})$ ii) $f$ is completely multiplicative $\Longleftrightarrow f^{-1}(p^{\alpha}) = 0; \forall p $ prime $\alpha ...
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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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Is the totient function $\varphi$ invertible?

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