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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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 that $g(n)$ is a multiplicative function as well

Suppose that $f(n)$ is any multiplicative function , and define a new function as $g(n) = f(d_2) + f(d_2) -...+f(d_r)$, 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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23 views

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 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 ...
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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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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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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 ...
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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 ...
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362 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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How to prove if an arithmetic function is multiplicative?

I know that for an arithmetic function to be multiplicative then $f(nm)=f(n)f(m)$ for $(n,m)=1$ I have just proved that: $$f(n) = \left\{ \begin{array}{l l} 0 & \quad \text{if 10|n}\\ ...
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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, $\alpha\geq1$. ...
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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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3D tensor multiplied with 2D Matrix

I have a nolinear system with it's taylor aproximation up to the second order so that: $\tilde{\omega} = f(\omega) : f = \sum_{j=1}^6 R_{ij}\omega_j + \sum_{j,k=1}^6 T_{ijk}\omega_j\omega_k + \dots ...
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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 $1$?
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Schemmel Totient Functions in Literature

I know how to prove that the Schemmel Totient functions are multiplicative, but I was wondering if someone could give me a reference to a place in the literature where such a proof is given.
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Multiplicative inverses and co-primes

I'm working out some examples on multiplicative inverses. I understand how to solve for a multiplicative inverse using the Extended Euler's algorithm, but I don't understand the principles which ...
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Summatory Function $F(n) = 1 $ for all $n$ odd, and $F(n) = 2$ for all n even

So, I have this summatory function $$ F(n)=\sum_{d\mid n}f(n)$$ that goes $F(n) = 1$ for $2\nmid n$, and $F(n)=2$ for $2\mid n$. This summatory function is multiplicative. I need to describe the ...
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168 views

Prove that if d = gcd(m,n) then $\phi(mn)=\phi(m)*\phi(n)/d$ [duplicate]

So if m and n are relatively prime, then the $\phi(mn)=\phi(m)*\phi(n)$ but what happens when $d > 1$?
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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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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 ...
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$g(x) = \sum_{m=1}^{\infty}f(mx)$ if and only if $f(x) = \sum_{m=1}^{\infty}\mu(m)g(mx)$

This is Problem 1.1.10 from book Problems in analytic number theory by Ram Murthy. It says, given the condition $$ \sum_{k=1}^{\infty} d_3(k)|f(kx)| < \infty $$ where $d_3(k)$ denotes the number of ...
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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) ...
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Conjecture related to the Erdős discrepancy problem

Conjecture: If $k \in \mathbb{N}$ and $S$ is an infinite set of primes, then the multiplicative $\pm$-sequence generated by $S$ contains $+^k$ as a substring infinitely often. (If $S$ is allowed to ...
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Multiplicative inverse of polynomial

Question: Determine the multiplicative inverse of $x^2 + 1$ in $GF(2^4)$ with $$m(x) = x^4 + x + 1.$$ My confusion is over the $GF (2^4)$.
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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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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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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 ...
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Figuring out a factor of modulo multiplication knowing other factors

So the problem is this - we have a simple equation: (A * B) % N = X All numbers are large integers. We know B, N and X, is it possible for us to figure out the last factor A without checking every ...
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How to calculate $2^{mn-1}/(2^n-1) \bmod{(10^9+7)}$

I was trying to solve Magical Five problem on codeforces. I have correctly formed an equation which I need to solve via program such that resulting number don't overflow. Answer can be Python or C++ ...
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How to show that the partial derivatives exist [on hold]

In general , how to show that the partial derivatives of a multivariable function exists without comupting it .
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$ g(n)= \sum_{d|n}\frac{\phi(d)}d=?$

how to find: $$f(n)=\sum_{d|n} d \phi(d)=? $$, $$ g(n)= \sum_{d|n}\frac{\phi(d)}d=?$$
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$\tau(n)\phi(n)\ge n$

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

how to find $$\sum_{d|n}(-1)^{\frac nd}\phi(d)=?$$
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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 $n$
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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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Composing Morphisms with Morphisms

Prove that the result of any such nested composition is independent of the placement of the parentheses. So this is what I have so far. Proof by induction I want to show that for any such choice for ...
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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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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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552 views

inverse,multiplicative inverse and Congruence of a prime field

I am dealing with ECC in these days which heavily based on finite fields. I want to how to find a inverse of a value in finite field and what is multiplicative inverse and also Congruence F29- ...
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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 ...
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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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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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182 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.